How do I increase the ios setting from 9.3.5 to at least 10.3 on an old iPad 2?

Sorry, not possible.

iPad2, iPad3 and original iPad mini cannot be updated beyond iOS 9.3.5 (WiFi Only models) or iOS 9.3.6 (WiFi & Cellular).

Apple ended update support for these devices in September 2016. None of these models can be updated to iOS 10 or later versions, as the internal hardware does not meet the minimum technical requirements for new versions of iOS/iPadOS. The 32-bit platform upon which they are based is effectively obsolete; later versions of iOS are designed to run on 64-bit hardware.

Many App Developers have also dropped support for older devices and iOS versions preceding iOS11/12/13 - and have withdrawn older versions of their Apps from the Apple App Store, now limiting utility and usefulness of older devices. 

If Apps that you need to use now have higher minimum iOS-version requirements, your only option is to replace your iPad with newer model. Of current models, even the entry-level iPad7 model supports current iPadOS 13.x and soon to be released iPadOS14 - and has hardware specifications and capabilities that vastly outperform your old iPad.

I hope this information is helpful - if only in clearly stating the limitations of your current iPad and your available options.

An iPad 2 isn't capable of running 10 or above.

(183684)

You cannot. The iPad 2 was released March 2011, it simply cannot by upgraded further.

Please buy a new, current-gen iPad.