Do you happen to have any hydrocortisone (no painkillers added)?
I read a thread about a chick who had a vent prolapse and after applying the hydrocortisone and resetting the prolapse for a few days it went away.
I'm sorry, I have no idea what the white gel stuff is that's leaking out. Does it smell bad or anything? What's the viscosity like? Egg white, mucous, maple syrup?
As far as sticking your finger in, I have no experience, but I'd say lubricate it with something and go very slowly and gently. Feel if you hit anything hard or sharp like an egg/broken egg.