If neither of you wants or can afford to keep the house, then it makes sense to sell it. Owning a house with an ex partner is not a good idea. Do not keep it as an investment or "until the market improves". Since you are unmarried joint owners, you will get 50% of the proceeds each.
The mortgage, you are not equally liable. Well you are, but I don''t think in the way you mean. You are "jointly and severally" liable. That does not mean you are responsible for 50% each... it means you are BOTH liable for the WHOLE amount. If the mortgage does not get paid then the lender can choose to chase EITHER one of you, at their discression. If they choose to chase you, then telling them "but I paid my half, chase her for her half" will not work, since you are both liable for the whole amount. Not half each.
This is why it''s a good idea to sell the house!!!
As for belongings, it''s a civil matter. If you can''t get her to give them to you voluntarily (or swap for some of the things you have, or pay her for them etc) then you will have to take it to court. And taking it to court is probably not a good idea. House contents are only valued at second-hand value, and the cost of taking it to court will most likely be greater than the value of the items.