My guess is that you either have a 1G or 2G iPod.
The 1G can only go to iOS 3.1.3 and the 2G to iOS 4.2.1. Kik now requires iOS 4.3 or later.
Your only way is if you have the version that was compatible on your computer such as in the recycle bin/trash or in a computer file backup. In that case delete the current version that is in your iTunes App library and replace it with the older version. Then sync.
What happens when you try to redownload it directly to the iPod by:
Do you have a Restrictions set that wold prohibit installing this app?