If you know for sure where to drill the base plate, do that, then you can push a pair of cables up into the wall. Have a helper push cables; with some maneuvering and pre-bending of the cables you might be able to catch them with a hooked tool (even another piece of wire) and guide them into the box, using that small space between the box and drywall.
If doing this with the wires doesn't work out, consider investing in a fish tape. Run the tape up the wall, snake it into the box, attach your cables, then with your helper pull the cables back down.
Looking at the photo, you may encounter another problem: the box may not be deep enough to hold all that stuff and the switch. Consider putting in a deeper box. In this case, cut away the old box and pull the exiting cables out, which will give you a big enough hole to get your hand in. Breaking out the old box isn't too difficult if it's an old Bakelite box; if it's plastic you'll need to nip away at it with cutters to free the wires, then pry it out. With the box out you now have enough room to get your hand on your new cables as your helper pushes them up the wall.
Install an old work box, which doesn't need to be attached to the stud but instead clamps onto the drywall. First, thread the cables into the box, then push box into the wall. Tighten the box screws, make up your connections and you're done.
If you don't mind patching drywall, cut away a piece about 12" high (4" above and below the box, plus 4" for the box) and width between the studs (16"). On the right side, cut to the center of the stud; on the left cut on the center except for the part next to the box.
This will give you full access to the box. If you find that you need to get a bigger box, you have more room to easily pop out the old one and install a new 'new work' (nailed to the stud) box.
To make the cuts use straight-edge and a utility knife first to score the drywall, then finish the cut with a small, thin blade saw. Do this carefully and save that piece; you're going to put it back later.
Fish your cables and if needed put in the bigger box (make sure you dry-fit with your removed piece.) Re-install the drywall with screws onto the studs, patch with compound and fiberglass tape and finish.
A tip: you can cover up less-than perfect drywall around a box by using a larger cover plate: either a "medium" or "jumbo".