rkresge has the plan: use a studfinder to find hardpoints within the wall and fasten the handles there. Just be sure not to drill into any plumbing or electrical in the wall in that spot. If there are no studs in the right places, consider spanning the distance with a piece of ply of solid wood (pre-finished shelving will also work if appearance is important) then mount the grab bar where needed.
My wife also has mobility issues and I mounted 3 18" grab handles to the front and entry door side of the galley cabinet. I used stainless steel screws, nuts, and washers to ensure a solid mount into the cabinet framing members.
If you need one in the bathroom, consider the front of the cabinetry or close to the door where there is framing behind the paneling.