Update to eslint 10 #4469

Is your feature request related to a problem? Please describe.
It'd be nice to update eslint to the latest version to get all the latest features and bug fixes and to allow us to use a newer version of typescript (the current setup only supports typescript up to 5.2.0).

This requires migrating the eslint config to the new flat config style: https://eslint.org/docs/latest/use/configure/migration-guide

The TypeScript plugin for eslint also has a different method of setting it up, requiring a different npm module to be installed (see https://typescript-eslint.io/getting-started/ for instructions)

Describe the solution you'd like
Migrate to eslint 10

Describe alternatives you've considered
Keep current version, but this limits what version of TypeScript we can use

Additional context
Upgrading TypeScript might be required to upgrade to the latest version of the languageclient (I'm not too sure at this point, I need to dig further into it)