I have sort of been on both sides of this issue. I understand how it feels for both parties involved and it sucks all round. I'll share what happened in case there's anything that can be applied to your situation.
I moved in with my then boyfriend when I was 16, I tried to go to college twice and dropped out, I could never get a job/hold down a job, was depressed, never left the house, no friends, didn't speak to my family etc etc. I hadn't yet figured out I was trans, I just knew I was miserable and unable to function properly.
Obviously this was very frustrating for my other half who was working and supporting me. This went on for years, we were unhappy but it was all we knew and there was still some vague glimmer of "it'll get better" - we'd make plans for a better life and never do anything to put them in to motion.
I tried to leave and moved in with my sister. It was very tense, she didn't want me there and the same issues came up of me getting a job and doing something with my life. He got in contact any way he could to beg me to come home and put out suggestions of suicide.
I relented and moved back in after about a month. Nothing changed and about 6 months after that I decided to write him a long letter basically explaining that I didn't want to argue or talk about the issue anymore, but the fact is I no longer love him and he clearly isn't happy either. I asked if it would be ok for us to break up, but still live together. I promised to do my absolute best to get a job and move out. In the mean time I agreed to take on full time voluntary work so I would be out of the house and building up some experience to help me get a paid job.
He reluctantly agreed to this, and there were occasions where he'd try to beg or reconcile. I felt heartless, but I deflected it the best I could. I eventually did get a paid job and moved out.
The attempted emotional blackmail continued from him, but I stayed strong, and he eventually got over it. We're good friends now, the past is never mentioned. There was a point a year and a half or so after I initially moved out that I ended up homeless - I moved back in with this guy for a while. By that point all the bitterness and weirdness had mostly subsided, I had started my transition and was starting to get my life together so we were just like normal guy room mates.
I want to add as an aside, in response to the comment that what he's doing is abuse. It is, absolutely, emotional blackmail. However I wouldn't go so far as to call it abuse - you have to look at the other side, it's a survival mechanism kicking in because he's afraid of being alone and fending for himself. You just have to decide at what point you say "it's not my problem" - there's only so much a person can do to help someone, you can bring a horse to water but you can't make it drink. If he refuses to get professional help for his problems, and can't/won't fix them by himself, there's not much more you can do short of having him committed and put on suicide watch.
Personally, if I were in your situation I would issue an ultimatum. As gently as you can, explain that you do love him but something has got to give and he's going to have to understand that it isn't fair to make you live like this. So he's going to have to get himself together (therapy, a job, voluntary work, school, SOMETHING to contribute to changing things) or move out. I agree with your boss, two months sounds like a reasonable time frame to me.