I had a bum leech of a roommate in college who was making life difficult while not paying rent and refusing to get a job and refusing to move out. I still have an urge to thwack something hard over the head when I think of that asshole.
Anyway, we were eventually successful in getting rid of him with no repercussions for us. I'll give you my tips:
1. Have a straight up, frank conversation with the roommate where you need to get across 2 stern things. Be unemotional and cultivate an air of competency and sternness. 1) The agreement is you would pay $X when you moved. You are not holding up your end of the bargain so our agreement is not valid anymore. (you can show her evidence if you want) 2) You WILL move out by Y date (not "you should" or "you need to"... "you WILL". Be firm!). Choose a date about 2-3 weeks from now (if you don't think she will trash the place) to give her enough time to line something else up. DO NOT allow her to argue or manipulate you. State the deadline date, and act like it is all but set in stone and all that is left is to negotiate how she will easily move on to her next house.
We actually printed up an agreement for the mooch to sign that said the same things. "NAME agreed to pay $X towards rent of ADDRESS along with roommates. As of TODAY'S DATE, NAME has missed $ABC in payments. This is a breach of our agreement. NAME agrees to move out on Y date. ______ signed (both us and mooch)."
This is a very difficult conversation to have. I sympathize. But you should get pissed off at this nasty mooch and channel that energy to getting her the f*ck out of your house and shovel that shit right out of your life. Being depressed doesn't give her license to be a gigantic mooching asshole. Plenty of other people with depression manage to be decent people and letting her walk all over you isn't doing her any favors. Hold her accountable, be pissed, and give her a necessary life lesson and kick her out. Use your words to make your displeasure (not holding up her side of the agreement) and her consequences (do not get to live here any longer) crystal clear. She is an adult and therefore beholden to adult consequences to not paying for the place you are living. You enabling her to continue in her funk is actively harming her, not helping her. She hasn't done anything productive with all this grace time you have extended her, and it's pretty clear she will not if you continue. Time to push her out of the nest.
2. Prepare yourself for her to enlist all your mutual friends and anyone in her corner to harass you or bother you on her behalf, and she will likely massage reality to cast you in a painfully negative light. Part of their willingness to go to bat for her will come from their not wanting to deal with her or support her themselves, and they were all perfectly comfortable while foisting all the shit onto your shoulders. Hold absolutely firm because you should know you are in the right. Rehearse a statement you can use to illuminate the reality of the situation to her minions and refuse to engage. Something like "I'm sorry you feel that way. I agreed to allow Susan to live there while she could pay rent, but she has not paid in X months. I cannot afford the rent by myself. Therefore Susan and I agreed that she needs to move out. (say this with complete sincerity--she knew the agreement and she broke it, therefore by the transitive property of not paying her rent, she agreed tacitly that she wouldn't live there). Oh, sorry, got to run, just saw Patrick and wanted to chat with him. Toodles!"
3. ENLIST THE LANDLORD as your ally! Let them know what has happened, and fess up to your mistake. Let them know you did it in good faith and vetted your roommate, but after moving in, circumstances changed and now the landlord should know what is happening on the property, and that you have a plan to solve the problem and here are the steps you are taking to solve both yours and the landlords problem(s), and if you had the landlord in your corner that would be very helpful.
In my case, we let the landlord know what was happening with our mooch-roommate. Because we had a good relationship with him (paid rent on time and didn't trash his house), he offered a last-ditch option in case the roomie refused to move off his property. That option was essentially formally "evicting" all of us, and then immediately amending the eviction and allowing the non-mooch roommates to re-sign a lease. That would have been a LOT of hassle and craziness so I'm glad we didn't have to go that route. But it was an effective threat to the mooch, and he did end up moving out a week after our "deadline".
The landlord will likely be an excellent resource on laws and such, entirely because they are a landlord and need to be on top of tenant laws and such. They may have a good read on if this will require the services of a lawyer if it gets beyond a certain point.
3. Once the hard move out deadline is stated, keep doing annoying check-ins. Every day. Without fail. Do not let her believe she can railroad you or "forget" to do the logistical work of moving out and then stay by default. This means you do have to take on some annoying logistical duties, but remember it is in your best interest in the long run. "Have you found a new place to live? You've got to be out by Monday, remember?" "Have you talked to your SO about how you will put together a security deposit?" "Did you go see that potential apartment like you said you would today?"
When things start going off the rails (and they will), start making noises around "well that is really unfortunate. Because no matter what, you will have to be gone on Y date. It will be pretty difficult for you to move out gracefully if you don't start doing those things."
And here is where the landlord would be a helpful ally. Because when she is sitting on her bum doing nothing to move out, the landlord can lend some intimidating weight behind you. They can act as a sort of "authority figure" that you may not embody to your roommate because you are around the same age and have known each other a while. That was a problem with our mooch, he didn't consider us an authority over him, but once we got the (older, more intimidating) landlord involved and coming around to check on his progress, he started hustling to get out.
4. I would also recommend speaking to a lawyer (perhaps your landlord has connections or suggestions) anyway, regardless of your landlords recommendations, just to CYA and see if there are any pitfalls you need to be aware of.