How do you properly address a letter to a lady?

Assuming you mean a woman with the title "Lady". On the envelope put "Lady Jane Doe" before the location address. For the salutation in the letter use "Dear Madam" or "Dear Lady Jane"

If you are talking about a woman who holds a position of power (e.g President) Use "President Jane Doe" on the envelope. "Dear Madam President" in the salutation

If you mean any woman, use Miss or Mrs. before her name on the envelope if you know her marital status, Ms. if don't (or if this is her preference). The salutation should be "Dear Miss/Mrs./Ms. Smith" if the letter is formal, "Dear Jane" if a friendly letter to someone with whom you are on a first name basis.