How can I reattach the piece and possibly make it stronger?
It looks to me as though the broken piece wasn't really attached to the fabric at all, or if it was, it wasn't attached very securely. If it had been glued with anything strong, the fabric would have pulled away from the sofa when the wooden piece broke off.
I would look at the rest of the wooden piece -- the part that's still attached. That part is probably attached to the sofa with nails or screws from the back side. If you flip the sofa over so that you can look at the bottom, you'll probably find that there's some black (or other dark-colored) material covering the bottom, probably just stapled on. You can remove those staples to free the fabric, which will then give you access to the inside of the sofa. Start with just the corner nearest the damage, and pull back enough of the fabric that you can see how the wood is attached.
The strategy should be to separate the wood from the sofa enough that you can glue the broken piece back on, and perhaps reinforce the repair from the back with a mending plate. You don't necessarily need to screw the mending plate on -- a strong epoxy like JB Weld would work well, especially if there are some holes in the plate for the epoxy to lock onto. Use regular wood glue (Titebond or similar) to glue the broken pieces together, and be sure to wipe up any glue that squeezes out of the joint with a damp rag before it sets up.