Same thing happened with mine.
You should try a SEUS update [free] or DaVinci [this costs]
If neither work it could be the flex ribbon. Other symptoms of this prior to being unable to switch on include the colours 'bleeding' [things turning into different colours] & the menu jolting as the icon animates.
You could get it repaired via your local phone retailers, but this could be costly. When I checked they tried to quote me £100.
http://www.mowtrad.com/ribbon_repair_services.html is a helpful website that could replace the flex & repair you phone for a smaller fee. This also includes a cleaning of the phones internal body.
Or, if youre feeling lucky, you could try replacing the ribbon yourself. I've found a company that retail the flex, along with a guide that should help.
Flex:
http://www.formymobile.co.uk/proddetail.php?prod=w850ribbon
Online Guide:
http://www.formymobile.co.uk/Dismantle/w850disassembly.htm
SugarKidd, December 2009