It's not a difficult fix at all. If you can remove the dial and hands, you can reset the keyless works.
With dial and hands removed, this is what you should see. Since you're working on a 5513 no-date, you may or may not have the datewheel installed under the dial; I've seen some instances where the datewheel is still present, even though the watch may be a no-date. So, I'll start with the datewheel:
Press this spring in to release tension on the datewheel teeth:
Lift the datewheel and slide it off. You should now see something like this:
Check these set levers to make sure they are engaged properly:
Chances are, one of these are out of alignment. A visual check should confirm that. Just re-adjust and you should be fine.
Install in reverse order, and always remember to only use a 1.2mm screwdriver tip to engage the release button for the stem on these!
Best of luck...