The L.E.D. ones don't really provide much in the way of useful light, from the ones that I've seen.
I couldn't find a flush mount light up mirror
I fitted a mirror that had two fluorescent tubes in. It was wasn't meant for flush fitting but the wall behind was batten and board, so there was enough room to mount it flush. The mirror sits flush against the wall but hanging on the original two brackets at the top (a). the weight of it keeps it flush to the wall and it's easily removed for maintenance (there were additional screws that went into side brackets of the mirror (b) but there is no access to these once the back is set into the wall) I'm happy with the result and it sits very firm. You could always run a bit of silicone along the bottom to fix it even more secure, but I found it wasn't necessary.
A problem you may have is the pull-cord. Mine is wired to the bathroom lighting cct on an occupancy sensor, so comes on when you walk in to the bathroom (negating the need for pull-cord).
I also replaced the fluorescent tubes with 12v / 12" cold cathode tubes (2). A more subtle light (easily enough room for the transformer) more efficient and last 10,000's of hours. The original fluorescents were 13w each BLINDING !
View media item 75391 = unused fluorescent starters
2 = new 12v 12" cold cathodes
3 = unused fluorescent connections
a = see below
b = see below
View media item 7540
a = top brackets sit into and over holes, weight pulling it down and flush.
b = bottom brackets, not used as no access to them
