Are you using the computer while installing updates? The updates sometimes pop up an installation dialog box that you need to click through to proceed with the install. Unfortunately, malware was exploiting pop-up dialog boxes, hoping you'd be typing something when the malware popped up an admin privilege elevation request, and you'd accidentally approve it. So Microsoft changed it a few years back so these pop-ups are now pop-unders. This means if you're doing other stuff while installing updates, the installation dialog boxes can show up underneath your other windows where you can't see them. The update just appears to hang.
So try closing all your open windows, or don't use the computer (have no windows open) while updating.
Some updates require other updates to be installed first before they can install successfully. Unfortunately, Microsoft doesn't seem to have a system in place to insure they're installed in the proper order. So if an update fails and there are other updates pending or available, go ahead and try installing them first. Reboot, check for more updates, install, reboot, etc. Eventually you'll get the required update installed, and the failing update will then install correctly. An alternative approach is to try installing updates one at a time, instead of everything at once.
You can try manually downloading and installing the update. A Google search for the failed update's ID will usually turn up a link to download the update directly from Microsoft's website. Then you can right-click it and run it as administrator to try installing it directly.
If you suspect Windows Update is borked, there's no need to reinstall Windows. Simply disable the Windows Update service. Then delete everything in the C:\Windows\SoftwareDistribution folder. Re-enable Windows Update (and manually do a check for new updates if you're impatient). That's the folder where Windows stores downloaded update, and maintains a database of installed updates. Deleting it will force Windows to check your PC for all updates which have been installed, then query Microsoft again to see which updates are available. Basically it puts Windows Update in the same state as a clean install (except you still have the old updates installed). Be forewarned that it will take a very long time the first time you check for new updates after doing this.