Sorry to hear about your external hard-drive. It's always a pain when a disk dies.
Although the distro you plan to install is larger than any flash drive you have, many distros do have very small network install images that can be installed on flash drives. As long you can get network access for the computer you are installing Linux onto, you can use the network installer to select the packages you want and they will be downloaded directly from the Internet by the installer. This large of a download will obviously take some time depending on the speed of your Internet connection, but it is probably the simplest way to get around your dilemma.
Another way would be to copy the entire distro onto your current computer and setup PXE boot on the computer you want to install Linux on, but this is extremely complicated, time-consuming, and frustrating. If you want to go this route (which I don't recommend unless you have a very high comfort level with bootstrapping already), google 'pxe boot installation' to get started.