I went through somewhat of a similar situation over the past 5 years. After 15 years the marriage was broken, no affection, a lot of bickering, etc.
She wanted to stay in the house so I rented a 1-bdrm apt nearby and for 4 years until our son went to college I provided monthly payments to cover most of the mortgage (it was in both of our names) and some living expenses. He spent 3 nights a week with me. As part of the divorce settlement she agreed to buy out my share of the equity of the house (about $300K) and take my name off of the deed. We had it appraised and she paid me by setting up a QRDO, funded with her retirement account. This does not sound like an option in your situation.
We both approached it like a business negotiation and that kept things from getting out of hand. We got along much better after I moved out and we are on good terms now that the divorce is final.
Continuing to live together if you are not sincerely trying to work things out is a big mistake. It will be confusing to the kids and stressful to both of you. She wants the divorce so she should make some sacrifices in her lifestyle or find someone else to support her. You will and should support your kids but absolutely hire a lawyer ASAP.
Since the house is in your name only (her mistake) just put it on the market. There is nothing she can do to stop you and it will force her to face the reality of the divorce. That will set everything in motion. The longer the separation and divorce drag out the more difficult it will be for everyone.