It has something to do with computer characters or a computer keyboard. Or the clues are translated into a string of characters that lead to some other answer.
Expanding on that:
It could be something like:
12 IZU 13 XX 4 $ (or ^2 $) 5 # -$ 13 ^ " ! < 10
The hint is the word "letters", which could refer to letters of the alphabet, computer characters, or communications sent through the mail.
Once each month I slept with you,
Could be 12 for 12 times a year or 30 for every 30 days. The second half could be the letters I and U or merely the letter U. "Slept" could indicate Z as used to denote a snoring sound. Also I notice that on the qwerty keyboard, the letters YUIO are all next to each other, which might be what "I slept with you" refers to.
But unfortunately I'm a young woman.
"Unfortunately" could indicate bad luck as in the number 13, and the young woman would have XX chromosomes.
I'll give it to you square: I'm rich!
4 is the number of sides of a square, and on a US English keyboard, the alternate symbol on the 4 key is $. Another possibility is the ^ symbol as used for exponents in many markup systems.
Halfway to perfect in an attack.
The repetition of "perfect(ion)" later in the puzzle makes me think there's some link between these lines, and this line is halfway through the riddle. Halfway to a perfect 10 would be 5. I'm not sure about "attack" but it could refer to ! ("bang") or maybe # (like a tic tack toe board). Also, "knowing is half the battle" - perhaps that's related.
My sister makes much less sense:
It might mean something that it's a sister and not a brother, father, or mother, etc. Could be a repeat of XX. "Sense" maybe is a pun on "cents" and therefore another reference to $. Perhaps "less sense" indicates a minus sign or less than symbol in front.
Her misfortune was to be created.
Bad luck and therefore 13 again. Then perhaps Insert or ^ for "created".
Twice I told her, I really did,
"I told her twice" makes me think of the double quote key or character. "I really did" isn't clear - maybe !.
But her perfection has now dimmed.
"Dimmed" could mean less as in minus sign or less than symbol. Again, "perfection" might indicate the number 10 or the corresponding keyboard keys.