refactor: reduce cognitive complexity of get_ai_analysis

[mohamedsolaiman]

Was looking through the AI analysis code and realized get_ai_analysis was doing too much in one function — HTTP error handling, SSE stream parsing, spinner management, all mixed together. The function had a cognitive complexity problem and also a bug where full_response was never actually populated (the return was always empty string).

Split into three helpers:

• _check_api_response() — validates the HTTP status and raises descriptive errors
• _stream_response() — parses the SSE stream, prints tokens, collects the full response text, and manages the spinner
• _build_request_payload() — builds the URL, headers, and payload

The main function is now just ~10 lines: build payload, make request, check response, stream response. Much easier to follow.

Also fixes the bug where full_response was never appended to, so the function always returned an empty string.

Closes #2581