Hello Janosch Lenzi,
I'm glad that you were able to resolve your issue and thank you for posting your solution so that others experiencing the same thing can easily reference this!
Since the Microsoft Q&A community has a policy that "The question author cannot accept their own answer. They can only accept answers by others ", I'll repost your solution in case you'd like to "Accept " the answer.
Issue: application insights suddenly stopped recording/showing collected data.
Solution: due to ingestion issues the application insights were not displaying however, it continued to record the data. After some time you are able to see the App insights automatically.
If you have any other questions or are still running into more issues, please let me know. Thank you again for your time and patience throughout this issue.
Please remember to "Accept the answer” and “upvote it", so that others in the community facing similar issues can easily find the solution.
Thanks