My daughter had an abdominal abcess off of her small intestine, (TI), connected by a small fistula, about 16 months ago. The treatment was the same drain line that you are recieving. They kept it in for about a week or so, flushing it regularly, and took it out when they were sure nothing was draining from it anymore.
She was on a large tapering dose of Prednisone as well. The Doctors at the time, also wanted to try a feeding tube for 6 weeks, but my Daughter dreaded that so much, that they decided it wouldn't be best for her.
After tapering the Prednisone, she began taking Imuran, and has been on that since. Her GI feels she has completely recovered at this point, (Pray). She even told us recently that it was not typical to have avoided surgery. If you have no fistula connected to the abcess, and no perforation, then the outlook for healing without surgery might be quite good.
Tough decisions for sure. I hope some of this helps. Thoughts and Prayers coming your way. Please keep posting updates, and Good Luck with you recovery.