Fujitsu Siemens and Dell Wireless 2350

Check another item first all wireless is not the same there are 4 bands I know of A, B, G, N. You need to check to see if there is one in common between your router and your wireless adapter,

I did not find your exact router name at Dell but a Truemobile 2300 is A/G and so is a Dell 2350 router (this is the right router they just call it two different names inconsistently on the site). Your new computer also supports B/G communications. OK here.

Did you configure the router or was it previously owned. The previous owner could have hidden the SSID. What this means when I did it with a Linksys was that I could see a router but it had no SSID (so I could not see the name) -- I had to enter my own SSID into the Wireless configuration for it to work.

I doubt this matters at this point but have you connected a lan port on the first router wan port of the second router. If so the lan side router address must be in a different subnet thatn your existing router -- if the existing router is 192.168.1.1 subnet mask 255.255.255.0 then all addresses starting 192.168.1.? belong to it and your second router must be different -- as in 192.168.2.1. OK Other Choice you only want 1 subnet so you connected a lan port to a lan port. Now the lan side of your router must be addressed in the first router's subnet statically
Assuming the first router is still 192.168.1.1 with a subnet of 255.255.255.0 and it has a DHCP server controlling addresses 192.168.1.100 for 50 leases to 192.168.1.149. Then we can assign uniquely 192.168.1.2 - 192.168.1.99 as static IP addresses to devices. Assign your Dell router lan address as 192.168.1.2 in this case (also change from a router to an access point - disabling the wan port). Also you should disable DHCP on this device as your primary router will handle all attached devices.

If we are not dealing with a hidden SSID (name) I would begin to suspect the router's performance. Unless as you review the configuration there is a way to turn the wireless radio off. I linksys equipment and cannot check that option directly, but I do not believe it exists on the linksys.
Check another item first all wireless is not the same there are 4 bands I know of A, B, G, N. You need to check to see if there is one in common between your router and your wireless adapter,

I did not find your exact router name at Dell but a Truemobile 2300 is A/G and so is a Dell 2350 router (this is the right router they just call it two different names inconsistently on the site). Your new computer also supports B/G communications. OK here.

Did you configure the router or was it previously owned. The previous owner could have hidden the SSID. What this means when I did it with a Linksys was that I could see a router but it had no SSID (so I could not see the name) -- I had to enter my own SSID into the Wireless configuration for it to work.

I doubt this matters at this point but have you connected a lan port on the first router wan port of the second router. If so the lan side router address must be in a different subnet thatn your existing router -- if the existing router is 192.168.1.1 subnet mask 255.255.255.0 then all addresses starting 192.168.1.? belong to it and your second router must be different -- as in 192.168.2.1. OK Other Choice you only want 1 subnet so you connected a lan port to a lan port. Now the lan side of your router must be addressed in the first router's subnet statically
Assuming the first router is still 192.168.1.1 with a subnet of 255.255.255.0 and it has a DHCP server controlling addresses 192.168.1.100 for 50 leases to 192.168.1.149. Then we can assign uniquely 192.168.1.2 - 192.168.1.99 as static IP addresses to devices. Assign your Dell router lan address as 192.168.1.2 in this case (also change from a router to an access point - disabling the wan port). Also you should disable DHCP on this device as your primary router will handle all attached devices.

If we are not dealing with a hidden SSID (name) I would begin to suspect the router's performance. Unless as you review the configuration there is a way to turn the wireless radio off. I linksys equipment and cannot check that option directly, but I do not believe it exists on the linksys.