The simple solution might be to use a really good grade COAX cable such as RG6 quad-shielded, or RG11. Then your attenuation should really not be too much of an issue. I use that to run a cell phone reception extender to an antenna in my attic and the loss is negligible.
However, since you've built yourself a can antenna, you probably already experimented with better feed coax, so here's the less simple and more expensive solution:
So, let me try to rephrase the desired topology... You've got the city signal that you can't receive very well. The soup can-tenna is doing the job well, but reception stinks from where your computer is. You have access to where the soup can gets good reception, maybe from the attic or roof or wherever better, but there's no power up there and you need power?
Do I have that right? One of the constraints is that there's no power but you need power where you can put this cantenna?
You can get a low-volt power source to do that.
You have a couple options for receiving the signal and getting it back to your place, but let me suggest the power solution -- use a Power Over Ethernet (POE) injector such as
this one or any of a zillion others.
Now, about the reception. You could get a wireless router that supports being powered from POE and stick that up into your attic, then put the wireless cantenna or a non-homemade directional such as a nice yagi onto its antenna port.
Here's one such POE access point. I didn't research whether its antenna can be unscrewed to put on a directional rather than the omni, but this might get you started.
If you can run an Ethernet cable from your computer to a place where the reception is good, then maybe the POE could solve the dilemma.