From easiest to hardest to do...
(1) Dont use xdcc's
(2) Use the other computer
(3) Copy mirc from the pc that works to the pc that doesnt, and use that copy
(3) Failing that reinstall the PC it dont work on
(4) Failing that Fix what ever is bung in ya router
Also define better what "kills your internet connection" means, to me that means all internet connectivity is lost for good, untill some manual activity is performed (router reset etc) to others it just means, while X is runnning Y and Z run like dogs, becuase X is eating all the bandwidth.
PS: Please be aware that if these are pirated files that your xdcc'ing from that for all we know you got a virus from one of your downloads which is the cuase of the problem, if so well ya get what ya pay for dont ya!