My wife has an iPad Air 2 is it worth replacing the battery?

Your iPad Air2 is listed as "vintage" by Apple. This indicates that Apple will continue to offer repairs or battery replacement while parts remain available.

However, as already outlined by Carol B, the iPad Air2 no longer receives system software updates...

The iPad Air2 cannot be updated beyond iPadOS 15.8.x, iPadOS 17.6.1 being current; iOS/iPadOS 18 is imminent. As many App Developers only support the current and immediately preceding one or perhaps two major versions of iOS/iPadOS, you might reasonably anticipate that Apps already installed on your iPad may soon themselves lose update support and/or may cease to work reliably.

Given that your Air2 is very much within its twilight of potential usefulness, you might be well advised to consider using the money that you might otherwise spend in replacing the battery in your existing iPad towards a much more recent iPad that remains in support.

An iPad battery replacement requires the sealed screen to be prised away from the iPad chassis - an operation that might itself cause damage to the screen or other internal components. If a repair or battery replacement is performed by anyone other than Apple or an Apple Authorised Service Provider (AASP), should anything go wrong you will have forfeited all support from Apple - and will be entirely reliant upon the third-party to resolve the matter.

This iPad was released about 10 years ago in 2014. Apple will replace that battery for a cost of $119. But Apple does not support iPadOS 16 due to hardware limitations so in the future, you may find some of the apps not working. If it were me, I would put the money towards a new iPad.

iPad Repair & Service - Apple Support