You wont find internet explorer in the add and remove programs simply because Microsoft have integrated it into the operating system, it is unfortunately a part of the desktop.
You could try an alternative browser like firefox from mozilla, but with no way to surf the web from that laptop, downloading it and transferring it over will be a complete pain, I suggest downloading it and burning it onto CD-Rom not a USB pen as that way you can install it off the CD without the dangers of whatever is thats causing you problems transferring itself onto the USB (Just in case its viral.) a free computer CD from down the newsagents with firefox on it might be an even quicker bet!
If it still refuses to work, you may want to consider trying a Linux live boot CD and boot the computer off of that and see if it suddenly springs into life.
Some anti-viral rescue CD's wouldn't go amiss either;
http://www.techmixer.com/repair-and-rescue-windows-os-using-trinity-rescue-kit-live-cd/Has it got anything installed like a firewall or anything that could be interfering with the normal operations of the browser? Sometimes a mis-configured firewall setting can be the cause of all the woes!
Sometimes people install crazy security configurations on their machines without meaning too and when they go to access the internet, they see the firewall pop-up with something about port 53 (DNS) and hit always deny thinking it's evil hackers trying to get into their machine, when in fact its the domain name resolution host and they've just gone and blocked it without a second thought.
