Never have an emotional attachment to a house!
Both design involve substantial structural change to the left front part of the house. When you get a quote you will realize it.
What I suggest is to buid a very low gradient sloped roof over the existing balcony and retain the railings, so that the appearance still about the same. This way the slow slope (I would say a 12/2 slope will be ok for water drainage) will prevent water from accumulating on the roof and get rid of the leak problem. Flat roofs alway are vulnerable for such leaks.
