Joints tenants or tenants in common require both names to be on the deeds.
If her name isn't on the deeds and you are not married and don't have kids together, she is king-sized, fúcked on making a claim for a percentage of the house.
Your MiL, if she can demonstrate, directly, a link of cash given for a deposit on the house on her behalf, she may have a claim for that.
I'd use the offer of giving her share of the deposit back on sale of the house as an incentive to get this over quickly.
Then fúck her out of the window for ever.