Is it permissible to take money mixed with lawful and unlawful sources from a woman who inherited it, with the intention of returning it to her, and is it considered a sin to cut off the unlawful relationship with her and limit the interaction to financial dealings for the purpose of recovering the money?

You must hasten to repent to Allah from the forbidden relationship, for fornication (zina) and its مقدمات (preludes/precursors) are among the most heinous sins. Repentance is achieved by abandoning the sin, regretting it, and resolving not to return to it, along with avoiding the causes of disobedience and cutting off the paths leading to it. The money must be returned to the woman and the relationship with her must be severed immediately. Beware of Satan. You must turn to Allah, avoid His disobedience, and rely upon Him, for whoever fears Allah - He will make for him a way out and provide for him from where he does not expect.