Ah, I see! Up to now you said the desktop remains connected indefinitely.
When this happens, can you access to router itself (not the Net) from the desktop, via its web interface, and post its Status and Connected LAN Devices?
1999: Freeserve 48K Dial-Up => 2005: Wanadoo 1 Meg BB => 2007: Orange 2 Meg BB => 2008: Orange 8 Meg LLU BB => 2010: Orange 16 Meg LLU BB