Contact a solicitor and arrange for the court to make an order for division and sale. Either your partner buys your share of the property, or it gets sold on the open market. There is no cheap easy fix though.
Your mortgage is simply a loan that is secured on your home. You own a half share of the asset and a half share of the debt secured on that asset. If there is equity you'll have some money in your pocket after the sale. If there is negative equity, you still owe the lender money as you will not receive a sufficient amount to pay off your share of what you owe the lender.
The date of valuation will most likely be the date when you separated. That means if the valuation has dropped (a strong possibility in the current financial climate) you will be in a better position than your ex-partner when the property is sold. If the property has increased in value you only get the half share of the valuation at the date of separation and your ex-partner will benefit.
Your solicitor should be able to help you be rehoused. You are homeless and not intentionally so. The local authority are obliged to rehouse you - especially given that you have a daughter. You could try talking to the council again as I suspect that whoever advised you previously has made a mistake.