Maybe the only answer is to air-gap the two devices and have the user input a secret code shown on the dongle. So you press the button, type the code on the website and log in that way.

Edit: Or maybe have two devices: one dumb flash storage and a device that actually computes your key. You press the "compute my key" button while the storage is connected to the keychain, then disconnect it from the keychain and connect it to the pc. Next time you connect it to the keychain, it's wiped before use and never ever read.