Subfloor Repair Glue

Repairing the famous soft spot in the floor in our bathroom. About a 3' x2' piece of Luan plywood. In this 2007 Rockwood 2607 it has a layer of luan laying across aluminum beams then another piece of luan that is set between the floor beams. I am just replaceable the top piece as the lower luan seems strong and stain free. I notice there is NO glue holding the 2 luan pieces together. Virtually no glue holding down the sheet vinyl floor covering.

I am not sure what holds everything down and together. Any suggestions on inserting a 2 x 3 patch into the floor system on this one?

Any help is always appreciated.