OK some questions for you, in order to better understand your needs:
Must this person be only in a relationship with you? Would they be free to have other outside partners, if the right person came along?
You said that the person must have a "connection" with both - are they expected to have an intimate relationship with both of you (i.e. do they need to be bisexual) for this to work? or would a close friendship with the one and a romantic relationship with the other be fine?
You mentioned them living with you - how soon after meeting a person would you imagine this person moving in?
Must the person be female? You wrote "lady" in the title, but would the right guy fit the bill?
"Listen, or your tongue will make you deaf." - Native American Proverb