Subtract $5M for rookie pool, which puts it at about $45M. $7M for Walker, makes $38M. I know they like to have some left for late free agent window signings and in-season spending, so I would assume that is close to $10M, leaving $28M.
Then, like you said, they still have some room for other contracts. They can renegotiate Jackson's and shave off $3M. Cutting Whitehair saves $5.8M, but you create another hole. I assume they will renegotiate his to get it down a couple of million as well. Vildor would be a straight up cut saving $2.7M, so those 3 could save $8M, pushing the available cap to $35M again very easily.
The best part is the Bears still have the highest available cap space after all those moves, $6M more than ATL and $17M more than the 3rd team (ARZ). So they are not done, I still think they get 2x more on the DL and another on the OL. They have double the cap space than 26 teams.