Client library
An associate developer has finally built this library:
https://gitlab.com/hectorjsmith/jira-api-client
There's a README on that site for how to use it. It builds upon the raw library - see below.
Raw Library Using OpenAPI
Thanks to s. hesse for pointing me in the right direction, I got a Kotlin library generated using the Open API tools. The Open API generator was a bit buggy, first complaining about the JSON provided by Jira and then creating Kotlin with a little bug in it. But with some tweaking the Kotlin compiled to a JAR. I haven't tested yet but here's the repo for anybody interested: https://github.com/ColmBhandal/KotlinJiraCloudClient.
Notes
The library that's autogenerated by the Open API Generator is pretty ugly. That's to be expected of auto-generated code from a tool that's not specifically tailored towards Kotlin. However, there's not much point prettifying it as the APIs may change and so regenerating this might be necessary. Instead, it's advisable to generate another layer of code between this ugly library and your client code. The code would be a sort of decoration/facade on top of the raw exposed code that's auto-generated, wrapping it to give type-safety and better ease of use.
Python Alternative
It also looks like there's a Python library out there for connecting to Jira, though we didn't want to use Python in this case: https://pypi.org/project/jira-cloud-python/