Reinstalling the launcher should be as easy as uninstalling and installing again, but there are some caveats.
The launcher files are pretty small in size and should hopefully not take too long to download. However when uninstalling the launcher its possible that the game files, which are contained in the SharedCache folder, will also be removed. Obviously we don't want that to happen as it can be an inconvenience to have to download the whole game again when we just want to download the launcher.
The steps for reinstalling the launcher only can be found below.
Get the latest launcher
Download EVE Online.
Check the SharedCache folder
Locate the SharedCache in the installation folder, by default C:/EVE/SharedCache/, but may be in a different location if you changed it during your last install.
Open the SharedCache folder
Once you have located the SharedCache folder you will want to move it manually as uninstalling the launcher will also remove the SharedCache files. So, move the SharedCache folder to another location on your hard drive before continuing.
Uninstall
Go to Control Panel > Programs and Features, or alternatively Windows Settings > Apps, find EVE Online and press Uninstall. Don't worry, your client settings will not be wiped by this operation.
Reinstall
Move the SharedCache back to the same location as before and run the EVE Launcher file that you downloaded earlier. It will ask you right away where you want to install the SharedCache, and should remember which location was chosen last time. So, if you have moved it back to the right place, just click next and it should be ready right away.
Start the launcher
Now you should be able to run the launcher again, with all downloaded files intact.