How to repair peeling corner bead?
Based on the overall condition of this outside corner:

The peeling of the joint compound appears caused by the corner being repeatedly hit.  

The corner bead does not appear particularly damaged, deformed, or detached. One could drive a 1-1/4" to 1-1/2" drywall screw through it without pre-drilling (a 10d nail would be sufficient to create a "starter hole", if necessary).

There's not much point in making repairs on this corner unless something is done to prevent further damage (ie. install plastic/rubber baseboard and plastic/metal corner guards).