Wow. Sounds tough.
I'm not really sure what to tell you. If yours was a 'relationship' question, it'd be a lot easier; the fact that it's more of a business relationship throws a few curves into the mix.
Okay, here's what I think.
If you don't already have a Contract together, then you need to make one.
I don't mean a contract regarding the rent, I mean a Contract regarding the labor, duties and expenses involved in sharing a house.
In this Contract, you need to address everything that's an issue: dirty dishes, cooking, cleaning the bathroom, using the telephone, sleepover guests ... whatever has been an issue or may become an issue is what must be spelled out.
So for example, your Contract might say that she will clean the bathroom at least once per week on every first and third weeks of the month; while you will do the same on every second and fourth weeks of the month.
It should also specify what the consequences are for not honoring the Contract (moving out? monetary fines?).
It must be made clear that the Contract (once made and signed by both of you) is non-negotiable and exists outside of any illness or circumstance (depression, break up with boyfriend, bad hair day, whatever).
This is so you both know the expectation is such that you will force yourself to honor the Contract no matter how you are feeling (that gets you over the 'I'm depressed so I can't...' attitude).
I don't know that making a Contract will help with the fight you're currently having (time will probably help that), but at least it might be a help for the future.
As far as the fight, I don't think you need to apologize although you could tell her your feelings and then let it go.
If you want to do that, I suggest you tell her something on the order of, "Look, I don't want to fight with you. It's clear that we don't agree about 'X' so I don't think we need to discuss it any longer. We're both still angry, so let's just leave it at that and agree to disagree."
Of course, that won't immediately get rid of the tension but it at least addresses the issue and might help clear the air somewhat. The rest will just take time.
I'm sorry I couldn't be of more help - good luck!