Your problem is the same one that I had with a Samsung SyncMaster 940BF monitor that my sister had given me to get repaired when it no longer awakened out of sleep mode on her computer (she decided to upgrade to an Acer 22" LCD monitor and gave it to me for helping her correct some other computer issues she was having). I gave it to my uncle to use on his wife's computer as it appeared to be at least partially functional (the same way yours is now), but she got frustrated with having to push the power button on and off repeatedly to get the monitor to display anything when she cold-booted the computer or awakened it out of sleep mode. So my uncle took it to the local "geek" in Stuttgart, KS, who diagnosed the problem as being the monitor's power supply board starting to fail. The "geek" told my uncle that the monitor wasn't worth fixing and tried to sell him a different monitor for $65. My uncle passed and took the monitor back to his place and let me know what the diagnosis was. I did some research and came across this site: http://www.lcdalternatives.com/index.html
I instructed my uncle to take the monitor back to my sister's house in Hays, KS and I would pick it up the next time I was in Hays. I then searched both the site and the associated eBay site and found the exact repair kit (the kit includes all the capacitors and a six-inch length of solder) and even purchased their soldering iron as well. After picking up the monitor from my sister's place in February (and after receiving the repair kit in the mail), I set about repairing the monitor and replacing all of the capacitors on the monitor's power supply board. After about two hours, I put the monitor back together and hooked it up to the DVI port on one of my desktop computers and powered both the computer and the monitor up and the monitor displayed a screen immediately and it stayed going. It has worked like a champ ever since.
Note: When looking for the exact repair kit for your LCD monitor, check out both their site and their eBay site. If you are unable to locate the exact kit you need for your monitor, you will need to open the monitor and remove the power supply board and give them the list of the board's capacitors and their exact specifications (in microfarads and operating voltages) and send them digital pictures of the board as well. If you do, you will get a 50% discount on the price of the kit for your "pioneering effort and contribution". It really is a cost effective solution to the problem you are describing. Let me know how it turns out.