It'd be easier to read the measurement on the tape if it was on top of the chain.
The current tyres are 26 x 2.0 (52x559).
The Race King 26 x 2.2 (59x559) should be 7mm wider in total or 3.5mm wider on each side.
The chain and derailleur look like they'll clear the bigger sized tyre but there won't be as much clearance as there currently is.
How much clearance is there with the chain stays and seat stays ?
Tell us more about your 'rebound' problem.