Are you on the most recent version? I was able to do this quest just fine on 5.3.2, and so have others, but for some reason people are still having this issue.
Seeing as I'm the only one who can fix it (or capable to do so) as of now, it takes me within a day to fix it. Although I was a bit busy with school so things have been slow. I'll fix it right now, though.
Inkay used to be a regular encounter there, but I erased it because it was too good of a Pokemon before Angie. The only place to get it in the badlands is in a place you're technically not supposed to be until at least v6.
Actually, I hid a way to purify your Pokemon at least once if you were so kind enough to test out Dr.Jenkel's machine. However, if you ask him to purify and you have nothing to purify, you lost your chance.