Your iPad2 is very old. Given your description, it is almost certain that battery has reached the end of its usable working life.
Battery replacement for an iPad2 is not available via Apple as hardware support ended many years ago. As such, for battery replacement, you’d have to consider a third-party repair shop - but before doing so, consider the following…
iPad2, iPad3 and iPad mini1 can only be updated to iOS 9.3.5 (WiFi Only models) or iOS 9.3.6 (WiFi & Cellular models).
Apple ended update support for these models in September 2016. Your iPad cannot be updated to iOS 10 or later major versions of iOS, as the internal hardware does not meet the minimum technical requirements for new versions of iOS/iPadOS. This limitation cannot be bypassed; these are 32-bit architecture devices, whilst all recent versions of iOS/iPadOS are designed for 64-bit hardware platforms. iPad2, iPad3 and iPad mini1 are all obsolete.
You need to consider that compatible Apps for your iPad are now unavailable. App Developers have dropped all support for 32-bit devices. As 32-bit versions of Apps have been removed from the Apple App Store, you’ll not be able to (re)download or install them - which may severely limit the usefulness and utility of your iPad. A battery replacement, if attempted, will completely erase the iPad of all data and installed Apps - the latter cannot be reinstalled.
As such, you would be best advised to save your money - and instead consider replacement of your iPad2 with a much newer or recent model of iPad. Of current models, even the entry-level iPad9 has specifications and capabilities that will massively outperform you current iPad - and is compatible with the vast majority of current Apps. Other models are more powerful still.
I hope this information provides succinct clarity as to your available options - and the fundamental limitations of your current iPad.
