Broken desk chair

[Gaben]

Hello guys, I have a little problem and I would like to know if any of you have any idea how to resolve it (besides my own idea). I got a chair, like the one shown in the photo, and today when i lean back the lean lock didn't ''activate'', and I heard like a crack from the chair's base (which the inside is made from plywood). Well now when I try to lean back all the lean mechanism doesn't work because It seems all the screws and stuff aren't screwed at all to the base because a little part of base cracked, and it doesn't have support. The question, it is possible to be repaired? My idea was like to cut that little broken zone, create a hole and introduce horizontally a piece of wood, screw it on both sides and then use it as a support for the lean mechanism, which is a piece of metal in L shape.