Johnye, the problem is the latest Kindle application require iOS 5.0 or later. You have iOS 4.3.5 which will not support the latest Kindle application. Your only option is, update your iPad to latest iOS.
You have to update your iPad from iTunes application on your computer. To update, download and install latest iTunes application on your PC from here : http://www.apple.com/itunes/download/
After installing iTunes, connect your iPad to your PC with the USB Cable. Open iTunes. iTunes will recognize your iPad. If iTunes prompted to Update, cancel or ignore it. Then select your iPad in the top right (left to iTunes Store button). Then under Summary tab, click Backup - make sure "To this Computer" is selected for backup. After backing up your iPad, click the Update button in there. This will update your iPad to latest iOS.
After updating your iTunes will ask you to setup your iPad as new or restore it from backup. Select "Restore it from backup". Then select your iPad's backup point from the list and hit Restore. This will restore all your data back to your iPad.
Then download Kindle application and see if that works or not.
If you have any trouble following the steps above, please let me know and I'll walk you through it.