If the monitor ever works at all, it's not the fuse. fuses once they
fail stay failed.

First make sure you're using the right AC adapter. It must supply the
SAME voltage as the panel needs, and at the same or more current. No
exceptions.

The most common cause of startup problems is worn out fluorescent tubes
(called CCFL tubes). How old are these monitors? If they've been ON
for more than a year or two, that's probably the problem.
Unfortunately replacing the tubes costs a LOT of money at a service
shop. If you try to do it yourself, it costs a lot less BUT unless you
are a skilled swiss watchmaker you're likely to break the tubes
(they're just 2mm in diameter and as long as the display). Don't even
think of doing this unless you're extremely skilled at electronic
repair!