I am trying to say 'I really do care and love this family'
I have also tried: 나는 정말로이 가족을 돌보고 사랑합니다
Tandem and google translate and naver all seem to have different ideas on how it should be spelt.
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It is a bit strange to say "I love you" when "you" refers to her family (not your own family) unless you have been living for a long time with her family. I would just say "I really care for and love your daughter," which can be translated as "저는 따님을 정말로 아끼고 사랑합니다."
As for the script, you should be more specific about whom you are referring to. If she lives with her parents only, you can say the following to her parents:
이미 아실지도 모르겠지만 저는 따님은 물론 아버님, 어머님과도 더 오래 함께하기를 원해 왔습니다. 그리고 따님을 얼마나 아끼는지, 이렇게 아름다운 따님을 훌륭하게 길러 주신 아버님, 어머님을 얼마나 소중히 생각하는지 말씀드리고 싶었습니다. 따님을 진심으로 사랑해 왔고 여전히 사랑하기에 이제 따님의 동반자가 되어 평생 동안 따님과 동고동락하며 살고 싶습니다. 저를 새 식구로 맞이하여 주시지 않겠습니까?
Because I do not know her family, I am not sure whether this expression will be appropriate; at least I can say that they will know what you mean. It is difficult for me to translate the whole paragraph, so I would just list most of the words I use above: