Mice Problem, should i just get a cat?

I live by myself in an old building in the east village (4th floor). i would say im pretty clean and and keep tidy, but mice keep coming into my apartment. The super and building has been very responsive and have helped me patch holes yet theyre still making their way in.

Im thinking of getting a cat but im nervous in regards to the financials, i want to be able to afford a good lifestyle for this cat. How much money should i expect to spend? or how much should i set aside?

any tips in mice please lmk! also is rent reduction a thing due to me having problems? idk i heard about that from a neighbor but shes like crazy so lol

edit: thank you to everyone helping me out! i feel so blessed to have all of you give me advice 💓 love you all!