Plan view
Corner detail
Length breakdown
Section through the door end. The cavity clears exactly one 2x6; the shelf's underside is its ceiling, which is why the shelf top sits at 6 1/4".
Short 1x8 floor boards run across the box, from a ledger on each long wall to a flat 2x6 spine down the middle. The spine rests on four supports set against your deck's joists: a full-width cross block just behind the fittings, a block on each of joists 2 and 3, and a cross-ledger on the far wall. Nothing is cut into the built box. Replaces the ledger-and-mid-span-beam scheme.
Plan — from above · floor seams dashed · your deck's joists green (solid measured, dashed assumed) · screws red
Long section down the middle, through your deck and joists — heights exaggerated ×2.2
Section across the box, 11" in from the door end — looking back toward the door
Cutting everything — four new 1x8s, and the 2x6, 2x4 and two 2x2s you already have