This is one of the most outrageous stories I have ever read on here!!!
You've had some good advice from others on what you could do as a next step. - But why isn't your solicitor advising you of these options?
Another option to consider/look into....
Could you just announce that you are moving back in? It is a house that you paid for. Why should a new boyfriend get to live there if you can't?? There may well be legalities which prevent you from doing so - but it is certainly something that I would look into.
You do need a mediator to sign to say mediation has failed or is not suitable. This is only valid for 4 months. It doesn't take long to sort out a mediation appointment. You should go and take Form.A with you to get it signed if she doesn't turn up.
Form A is what you use to apply to the court for financial settlement.
There is a cost for doing this but in the long run it could save you money.