Pull backrest of sofa forward toward the seat until you hear a clicking sound.
Try to lay the backrest flat until you hear it "clack" into place.
Jiggle the backrest slightly if it got stuck either before your heard the "click," or the "clack." Gently try to move it in the right direction.
Note the position where the backrest sticks. This will help identify the piece on the frame that needs to be adjusted.
Examine the sides and bottom of the couch to find the reclining mechanism. The poles that support the frame or the pins that hold them in place may be twisted in the wrong direction, or bent out of shape, preventing the seat from reclining. Straighten the pins or poles with a hammer or a wrench and attempt to move the seat into proper position.
Order a new part, if straightening the frame doesn't work. Furniture repair companies sell parts for convertible sofas. They can also offer advice on which part you need, if you describe the model and problem.