It could be one of many things, most of them unfortunately fatal and not something you can repair yourself (or get someone else to repair for much less than the cost of replacing the machine with something new). As the battery has expired I assume it is no longer under warranty, so you probably can't return it to the manufacturer and ask them to sort it.
On the off-chance that it is just the display (or the ribbon cable to it) that is malfunctioning, try plugging it into an external monitor.
If it can't be repaired and you don't have backups of your data all is not lost in that respect, assuming the drive is OK which from your description it probably is. These units, like most netbooks, use standard 2.5" drives so you can remove it and put it into a SATA->USB enclosure in order to read off the data (or just us it in-place) - and you now have a handy external drive that you can use as part of your backup strategy when you get a replacement the machine. Getting in to the drive isn't terribly easy but there are tutorials out there, like this one for example.
If you didn't have backups of your data, do make sure you arrange a good backup arrangement ASAP, it could save you a lot of stress in the future!