I have a ver 2.6 (as opposed to your 2.5) Linksys USB Wireless adapter. I don't know what the differences are; Linksys treats them differently in terms of drivers for sure.
I have a fix; it is sloppy right now, and I am developing a better one. The root cause is that the Webplayer doesn't supply enough power to the USB device. I investigated this quite thoroughly; for reference, I was using Win98 (should have no bearing on the problem if power supply was the cause) and no other devices on the other USB port.
The solution I found was to bypass the Micrel 2545A chip between the 5V (well, 5.24V as measured) supply and the USB ports. The M2545A is a current and thermal limiting device with an enable - basically used to both limit the total current drawn by both USB ports to 500 mA (or 1A total) and to allow the Super I/O controller to hard reset USB devices. I bypassed this device by lifting two pins and shorting the full 5V supply trace to what would otherwise be the current limited trace to the USB port. When the Webplayer powers up, the wireless connection link light comes on - as it did before I mucked around inside the webplayer - but doesn't provide network capability. However, if I unplug and plug back in the USB cable to the wireless adapter, network connectivity is achieved - I noticed during power up that power to the USB connector is severed several times by the Super I/O chip.
My solution right now is awful - I have to unplug the wireless adapter and plug it back in each time I boot. While the adapter is getting enough power - I have no idea how much, but I will measure it before I complete this project (for completeness) - it isn't getting reset at the appropriate times by the Super I/O controller. The current draw allowed by the M2545A chip is set by an external biasing resistor - R = 230 / Imax ; in this case, I measured an R (the resistor on board) of 230 ohms (the resitor is silkscreen marked with "R237" as an identifier (meanining it is resistor #237, not that it is 237 ohms). The Maxim spec sheet says that current deviation of +/- 30% at "low" (i.e. sub Imax=1A) current draw was possible. I suspect if I swap out this resistor for one of, say, 115 ohms, I may allow more current to be drawn from the USB ports. Of course, this is mildly dangerous, but not too bad.
I suspect this solution, if it works, may solve quite a few problems people have been having - including those that require a "powered hub" to solve. I plan to try it out this weekend - I am quite busy in my real capacity as slacker engineer right now - and will post both results and pictures to aid in modification if this turns out to work.