The main thing I would be worried about is the water detail where the joists come through the sidewall of the house. Also, are the existing joists pressure treated and are they showing any sign of water damage or rot? I have rarely seen cantilevered deck joists installed correctly. I have seen many cases where the flashing wasn't right and there was extensive rot to the sidewall structure. Do you have any sign of water leaks inside, just below where this deck comes through?
Basically, you risk flashing leaks and rotting out the side of your house if you keep the cantilevered joists. Any problem that develops could easily compromise your house. There is really no reason to incur this risk because you aren't going to be using the cantilever. I would remove enough siding to be able to sawzall the joists off flush with the sidewall framing. Then I would lag a ledger board/joist (pressure treated) to the side of your house, use regular joist hangers, and lead flashing over it and up and under your siding. If you are 100% certain that you have no leaks and that the existing joists have zero rot, then you can just sister joists as Dave said.