Update litterrobot documentation to match current implementation

[natekspencer]

Proposed change

Update litterrobot documentation to match current implementation

Type of change

• Spelling, grammar or other readability improvements (current branch).
• Adjusted missing or incorrect information in the current documentation (current branch).
• Added documentation for a new integration I'm adding to Home Assistant (next branch).
  • I've opened up a PR to add logo's and icons in Brands repository.
• Added documentation for a new feature I'm adding to Home Assistant (next branch).
• Removed stale or deprecated documentation.

Additional information

• Link to parent pull request in the codebase: Add last seen and status code diagnostic sensors to litterrobot core#71760
• Link to parent pull request in the Brands repository:
• This PR fixes or closes issue: fixes #

Checklist

• This PR uses the correct branch, based on one of the following:
  • I made a change to the existing documentation and used the current branch.
  • I made a change that is related to an upcoming version of Home Assistant and used the next branch.
• The documentation follows the Home Assistant documentation standards.

[probot-home-assistant]

It seems that this PR is targeted against an incorrect branch since it has a parent PR on one of our codebases. Documentation that needs to be updated for an upcoming release should target the next branch. Please change the target branch of this PR to next and rebase if needed.
_{^{(message by DocsTargetBranch)}}

[frenck]

@natekspencer Small reminder, the parent PR has been merged and released, but the documentation hasn't been updated yet (as there are comments open). Could you take a look?

[natekspencer]

@natekspencer Small reminder, the parent PR has been merged and released, but the documentation hasn't been updated yet (as there are comments open). Could you take a look?

Done, thanks for the reminder!