From what I've read, the only reason you still live together is that you can't afford to move.
That is, even when your bodies live together, your souls don't.
Look into yourself and find out whether your relationship can be recovered or not. If the answer is "no", then let her free. If both of you have decided to split, both of you have the right to start over again, and even when she told you she wouldn't start a new relationship, it doesn't mean she must stick to her word. Maybe she wanted not to hurt you.
When couples split, both parties need to feel they are still "in the market" and they can start over again. Women have an easier time finding a new couple, even if it is just "casual" or "temporary" and that way she feel atractive and desired again. Probably her relationship won't last long. Probably she wants you to be jealous and "fight" for her, and put your love to the test. Now it is up to you whether you really want to start over or you really want to recover her.
From my point of view, telling she won't date anyone and doing is a white lie. It is worst to live for years believing your other half loves you and she/he doesn't.