It took me ~4 months. I interviewed with a handful of really great companies and made it to the final round with 4. Companies are definitely willing to interview/hire self taught developers. LS career services overall were pretty useless, but there were a few staff members who helped give guidance and I am really grateful for that. Try and find someone in the field who can help mentor you.

I'm self taught like you, my biggest regret is not sending out resumes earlier. Try to find small and midsize businesses they are the most receptive in my experience.

Things might have changed since I finished the program, but their career services were effective for me.