Ok, there is a way to get those videos off. It is possible, but you are going to have to accept a few things in order to get things done.
The first is that you are on Windows XP, an operating system that Microsoft will be discontinuing support for soon, especially since Windows 7 is a big success and Windows 8 is on the way. There seems to be some kind of file size limit that has to do with your version of Windows and your file system. I have not run into any issues with transferring large videos from my iPad or iPhone on my Mac or my Windows 7 Machine, everything just comes over as smooth as butter.
Second, there isn't anything wrong with your iPad. bmike is right in what he is saying about the youtube file size limit when uploading from your iPad.
Now, the easiest way to get those videos off in my opinion, which you might not like, is to take your iPad over to a friends/family/coworkers house that has a computer running Windows 7 64bit or a Mac with Snow Leopard or later (Don't forget to bring your transfer cable too). You will be able to plug in the iPad into these more up-to-date OS having machines and use Windows Photo Transfer or iPhoto on the Mac. Also, you should bring a flash drive/external hard drive with you in order to leave with the video. And hey, no 3rd party software and its technically free.
Also, if you plug your iPad into your friends machine and iTunes opens, just close it, you won't be needing to use iTunes in order to do any of this.
I know it isn't the answer you are looking for and it technically should just work, but it seems that the biggest issue you are facing is that you are running an older OS. It sucks, I know, it should just plug and play and work but the OS is the thing giving you the problem. I have had similar issues with things that work perfect in Windows 7 just face planting for no apparent reason in Windows XP. This is the reason I upgraded all my machines to Windows 7 because it is the most up to date and receiving a lot of support. It fixed all the little drive issues and bugs that were happening with new peripherals/software/hardware on such an old OS. Windows XP is 11ish years old after all.
Also, I know this doesn't solve your problems in the future, if you take a huge video again, you will still have the same issue until you upgrade your OS or buy another machine.
I really hope that my answer helps you out even if it's not in the spirit of no additional tools in case no one else has a better solution for you.
You have informed me that you are working on a machine that is locked down by your job, a work computer from a company. This complicates things further and continues to make my answer one of the easiest and simplest answers for you.
You need to use a different machine that is running an up to date OS.
Your job might be locking something down and not allowing you to do something. I have done a lot of work with AT&T laptops that are super locked down for too many security reasons, yours may not be as locked down or secure but it might be one of the things stopping you besides the smaller file size limit from your OS and the fact that it is 11 years old.
I know, it sucks, you want it to work the way you want it too, but you aren't going to get it to do that. I can't put diesel fuel in a regular car. The engine will not work with that fuel. You need a different engine. You need a different computer. I am not saying to go out and spend money on a new machine but if you plan on doing things that are up to date and in the current mode of technology and things that are currently supported, I would highly recommend getting a home computer that is more updated and current. You iPad is a year old, the OS your trying to use with it is 11 years old. I am not trying to sound mean, I am just really trying to get my point across. I understand it sucks that your machine won't do what you want it to do but technology does that sometimes, it gets old and obsolete and new things don't work with the old stuff anymore. There isn't really a way around it.
I had a similar issue with a friend of mine who had a older macbook pro 2006. Its a great machine and ran games like WOW and others fine. But Diablo 3 will not work on it. He had to buy a brand new computer in order to have it work. He complained because the computer is only six years old. A lot can happen though in six years. Its just how technology is. There are always going to be certain things that have legacy support but not everything will and just because it says it does doesn't mean everything will always work or work 100% of the time.
So, in short. If you use a new computer to pull the video off, it should work fine. Your older work-locked-down computer with the outdated OS will not work with it. You will need to get a new machine eventually, so keep that in mind. Your older computer isn't going to work with a lot of new things going forward, especially when Windows 8 comes out.