How to fill an old window alcove?
Skip the studs - there's nothing structural here. 


Use (roughly) 9-10 cm of rigid foam insulation and 1-2 cm of plasterboard/drywall or plaster, depending on internal finish of the rest of the room.
Glue the foam in place with a construction adhesive meant to work with the type of foam you get (should come in a caulking tube.)
Use some canned polyurethane spray foam ("minimally expanding" type)
to fill in the edges around the rigid foam sheet, allow it to cure,
trim back any excess, and finish.