Live Update is not a real updater. What it does is download the ZIP package to your site's temp-directory, extract it, then tell Joomla!'s extensions installer to install the (extracted) package. It is possible that the server configuration didn't allow Live Update to download the package, the temp-directory is unwritable/incorrect or that your server does not support extracting ZIP archives. Half of these issues will be fixed if you follow the installation troubleshooting instructions. The rest have to do with server setup and can't be fixed by you, only your host.
As a side note, when I told you to follow the installation troubleshooting instructions I did that having in mind that this will most likely also fix the root cause of Live Update's failure. When you see me insisting on following some specific instructions (even if they sound irrelevant to your issue) I have thought of at least one reason why following them might actually solve your original issue. I just don't want to do the boring technical analysis of why it might do that. I think you mot likely want your problem solved instead of reading a dull lecture of mine full of technical jargon :)
Nicholas K. Dionysopoulos
Lead Developer and Director
🇬🇷Greek: native 🇬🇧English: excellent 🇫🇷French: basic • 🕐 My time zone is Europe / Athens
Please keep in mind my timezone and cultural differences when reading my replies. Thank you!