This works for Win 10 and may work for you but from the Safe Mode desktop, press CTRL+ALT+DEL then click on the Wifi icon then on your SSID and then on Connect - left click away to close that pane then click on Cancel, but if that is approx. what you are doing then it should connect.
In Win 8/8.1 and 10 in Safe Mode, Windows disables wireless so it's just as easy to wire up to get connectivity.
Can you go Start - start to type reliability and press enter when View reliability history comes up to see if that has recorded anything for the failed updates - click on the blue info link if any and that may give you an error code.
Before running the repair program can you open a Command Prompt (Admin) in normal mode and enter these cmds -
dism /online /cleanup-image /startcomponentcleanup
dism /online /cleanup-image /restorehealth
sfc /scannow
The first two should report as successful and the sfc /scannow is to check if any corruption remains.
If you still have update problems then perhaps you should opt for a repair install which won't affect your personal stuff or installed programs.
However, it's advised to back up important data before proceeding.
If you don't have Win 8.1 install media then you can download the ISO from
https://www.microsoft.com/en-gb/software-download/windows8 and use
https://www.microsoft.com/en-us/download/details.aspx?id=56485 to create either a bootable DVD or USB.
http://www.intowindows.com/how-to-repair-windows-8-1-installation/As updates are default installed during the process, it may resolve your update problem.