the logicboard seems to be ok , otherwise the external screen would be dead too , so all you need is to find a used but faulty iMac G5 where the display /inverterboard still works fine , and that can be had cheap , but still you are looking at around £100 at least here in the uk for a faulty iMac G5 , plus the work to install it ,
because you need in that early G5 no isight to dismantle the whole iMac tin order to get to the display /inverter board , you should best replace straight away all of caps without looking because they are little timebombs , good soldering skills needed , so if you dont have these skills you are looking here in the uk at about £170 , means
100 for a working display and inverter 170 for recapping makes about £270 , but the result would be a fully refurbished and ready for the next years to come iMac G5
or you could get a nice monitor ~19" for far less place it in front of the iMac and use the iMac as long as it takes you or your kids to save up for something else from apple,
but no matter which new "used" Mac you choose inform yourself really well about the major problems the Mac of choice had , just because most Mac's get sold after apple care has ended ,and apple is really tight when it comes to acknowledge a fault and offer repair schemes after apple care , you just need to know what cost can hit you with a sledgehammer , and the intel Mac's had not been without faults
the safest option for a used Mac is still the Mini , and as you need a external display anyway if you keep the iMac G5 until there is some cash saved up the mini is a logic option , the intel mini core solo 1.5 ghz will set you back about as much as it cost to repair the iMac G5 and offers a tiny bit more performance too ,
if your kids (dont know how old they are) are a bit into tech the core solo can easy and fairly cheap be upgraded to a slightly over 2ghz core duo i think 2.16ghz but dont know exactly)by just fitting a faster processor takes a half hour without knowledge, by just following instructions ready available on the web , this upgrade is often cheaper then buying a used 2 ghz mini !
with slightly less performance compared to the iMac G5 but a lot cheaper too you could get for each of your kids a eMac G4 1.42ghz the most underrated Mac of all times and nearly bulletproof , i have 3 of them and vouch for that Mac to last longer then your intel iMac even in kids hands , because thats just what it was build for , to be abused by generations of children
just forgot :
another thing your iMac G5 does not have a mini dvi port , it has a mini vga port
just something to remember is the iMac G5's max resolution support on external displays
exact info to yours you can find here
http://www.everymac.com/systems/apple/imac/index-imac.html
