Refer to the apartment as "my apartment" - even though Mrs XXX is the owner, you are the one living there and using all the facilities.
I'm not sure what you meant by "when I intended to use it after I moved into the house". It is clear that you would expect a telephone in your apartment to actually work.
"Yours sincerely" is overly formal. Given the brevity of the letter, I think "Yours" is sufficient.