Having now read your previous threads, the last being when your h 'boxed' your head and had taken himself before the arrival of the police and ambulance service, I would suggest, if you have not already done so, you act on the advice previously given to make contact with your nearest Women's Aid service which you can locate here: www.womensaid.org.uk which is available to all women regardless of nationality/status etc.
Regardless of the content of any tenancy agreement or other contract between your h and the landlord/owner of the property you currently reside in, you cannot be evicted without due process of law from your rented marital home which means that the landlord/owner must first give you notice to quit by a given date and, if you do not comply, they must serve notice on you of his/her intention to seek repossession of the property before instituting proceedings in the civil (county) Courts for an order to evict.
It can take 3/4 months, or longer, after the date any tenancy/contract comes to an end before a landlord/owner can lawfully take possession of the property.
NB if you are not married to the man you call your 'h' in your posts and are not named on the tenancy agreement/contract, you may have no legal right to occupy the premises and, if this is the case, I would again suggest you make contact with Women's Aid (as above) or make contact with Shelter www.shelter.org.uk
It is unrealistic for you to expect that drawing up a deed of legal separation will compel your h to support you and your dd as the fact is he will be required to agee to any terms contained in such a document and, as he's made it clear he does not intend to support you, it's improbable that he will change his mind.
Athough he can be compelled in law by the Child Support Agency or the Courts to pay a small percentage of his income towards the cost of maintaining your dd, he cannot be compelled to maintain you unless directed to do so by a Court of Law and, from what you've said, you have already been advised by a solicitor that neither of these scenarios are likely to happen within the next 20 days.
Did you pay to consult a solciitor or did you take advantage of a free half and hour initial consultation offered by some firms? What makes you think you are not entitled to claim State benefits?