Where is the best place to tap into this broken USB flash drive with a new cable?
Three of the four pads ripped off the board along with the connector. You will have to very carefully scrape the resist and connect to the vias for the D+ and D- lines as shown here:



The colors notated above are the wire colors you'll find in most USB cables if you cut off the "B" end and expose the wires.



EDIT: One way to make the resulting assembly a little more rugged would be to use another board as a strain relief for the cable. Hopefully, this sketch makes sense:



Get a piece of pad-per-hole board or even just perfboard, and tape the USB board to it, after putting something like a piece of cardboard or tape between the two boards to prevent shorts. Solder the wires from the USB cable to individual pads on the bigger board, and then use single strands of very fine wire to make the final connection to the USB board.

This way, any movement of the USB cable relative to the larger board won't put any strain at all on the delicate connections to the USB board.