I have to admit I’m not an expert on breeding, but I’ll definitely share my thoughts. My first thought was that your idea is an awesome project! My second thought was that I’m envious!
Bear with me- I’m thinking this through as I type it out. That almost always means I have run-on sentences and too many parentheses. Hehe. 
If the egg she previously laid was what I suspect- a collapsed egg, that means an egg that she could/did not immediately lay for some reason was crushed inside her uterine horn. The discharge from the vent would be the contents of the egg.
If it was only the egg that she laid that collapsed, and she just didn’t have a second egg in her other uterine horn because she’s a young female (was it intact or perforated at all?), then the question to be concerned with is what caused it. If she found her laying hide ‘inadequate’ for some reason (and this has happened to me before for no apparent reason, as well as because I accidentally walked in the room & made loud noises the night my girl was laying), she may have kept from laying it for too long & then experienced problems. The solution could be as simple as providing two quite different laying chambers and making sure to be extra quiet.
But if it happened for some other reason, it may be a greater cause for concern. If she was low on calcium or some other minerals (and I say that with zero judgment- I’ve done it), it could be that her muscle tone was too poor to lay without having to strain herself repeatedly. In that case, the solution would be to make sure she’s all amped up on supplementation and nutrition, which is an easy fix. I have zero idea if it’s a possibility that W & Y morphs can have neuro problems related to the contraction of their uterine muscles, I have not heard that, but it occurred to me as not an inconceivable explanation. Has anybody else heard anything like that?
What I’m most concerned about is if the collapsed egg/source of the goo is actually a second egg that she has not laid, and that the shell is still retained, unlaid, inside one of her uterine horns. I am actually dealing with this with one of my 19 year old geckos, Binky. I don’t know what would happen if an egg began to develop in the horn with the old egg shell still in it and then failed to be reabsorbed- I don’t know if a uterine horn can still function, or if the old egg shell would be stuck in place or blocking normal laying in some way.
I’m actually very worried about this for my gecko Binky, as I know from her radiographs that she has a calcified shape right in the spot where a collapsed egg would be. I’ll see if I can dig up a copy of her radiograph. Last breeding season she went off food for almost all breeding season, and I had to syringe feed her Grub Pie for several months & give her anti-inflammatory drugs (under a herp vet’s care). After the breeding season was over, she was her old self and eating like a little piglet again. My solution (because she is very elderly and not a great candidate for surgery) has been to put blackout drapes covering my new reptile room’s window, and creating an artificial 12/12 day/night cycle that doesn’t vary. I’m hoping this will help at least lower her hormonal response to the season change, and my herp vet thought it was a decent idea to try as well. I’m just going to have to watch her like a hawk for signs of egg binding, and probably feed her by syringe for a few months if it causes her to get uncomfortable again during this breeding season.
The good news is that, since your girl was able to expel the contents of the egg, I think it’s safe to assume that her uterine walls were not perforated, as she would have developed coelomitis & died. My gecko Steve-O died from a ruptured uterine horn following her surgery for egg binding, as the egg had gone unlaid for so long that she lost all muscle tone in one uterine horn. It wasn’t then able to drain normally through her vent, and when fluid built up, it ruptured through the horn and developed into sepsis, even though she was on antibiotics. (You don’t generally remove the uterus in a leopard gecko ‘spay,’ as they may have retained, microscopic egg follicles that get missed, and if they do, and there’s no uterus to lay the egg, they 100% will die if they don’t reabsorb or have surgery.)
So, I guess (I’m still thinking this through), if you wanted to know for sure if there is a collapsed egg shell still retained, a reptile vet could radiograph her for you & it would be pretty easy for them to tell. And, dude, let me tell you- a radiograph is a LOT cheaper than an egg-binding surgery. If there is no shell on the radiograph, I think it’s safe to assume that she only had one egg to start with, and that egg was definitely laid. In that case, I’d think it’s pretty likely that you could breed her successfully, with some pampering. If she does have a collapsed egg, I’d talk to a VERY experienced reptile vet to determine if she needs removal surgery first or just shouldn’t be bred at all.
That said, I understand that most people aren’t going to want to radiograph their gecko. In the event that you really want to try breeding her without a radiograph, I would take every possible measure you can to ensure her optimal health, choice of laying hides, and subdued external environment and watch her extremely closely for any signs of egg-binding. They are unfortunately more subtle than I would have thought, and pretty easy to confuse with laying behaviors.
My smaller females will sometimes go off food regularly when they have almost fully developed egg(s), then eat again immediately following laying, then go off again once their next eggs get large (~3/4 week cycle). Laying behavior also includes a lot of hanging out in the lay box and digging around. But my girls that were egg bound simply didn’t come out of their regular hides, at all. They wouldn’t eat. Major lethargy, but only one of them looked abnormally distended. Both were 15+ years old and had not been bred that year (obviously, heh).
Oh, I’m glad I’m re-reading, I had forgotten that she laid an egg afterwards! Depending on how long after it was, it could have been from the same clutch, or a separate clutch of one. My guess, because the first egg was in a bad state by the time she was able to get it out, is that it was the next cycle. But I have never seen drops of blood with laid eggs, and not mold, even on infertile eggs, unless I don’t discover them for days. Given how difficult the first egg was to lay, it seems pretty likely that at least one uterine horn had formed fibrin adhesions (major scarring) that made it physically traumatic to lay the second egg. Or, if it was from/in the other uterine horn, it could be that she had to squeeze the second egg past a retained collapsed egg.
I’m not sure what to make of the second egg she laid. I mean, on the one hand, it confirms that she was able to successfully lay an egg from one or the other uterine horn, but we don’t know which one, nor whether there’s a retained collapsed egg shell involved. I’d say that, ultimately, it’s sort of both good and bad, because it’s good that she successfully laid the second egg without seeming ill again, but it’s concerning that she had to strain/have tissue damage of some kind in order to lay it (extrapolating from the drop of blood).
Overall, I’d say my assessment stays about the same. Personally, I would get a radiograph before trying to breed her again (side benefit you’ll get a decent look at her bone density), but I understand that most people would not. If you do decide to try breeding her, I would just watch her like a hawk 
and be ready with the money to pay for emergency egg binding surgery or euthanasia.
I hope that helps. I’m really sorry it’s so long and that I’m so wordy, I promise it’s just my intent to help & not just to be super annoying, hehe.
