Rather than using $.ajax, I've used a simple image request with URL parameters for sending back error info. That has the benefit of allowing cross-domain transfers. Be sure to save the user-agent info as well so you can correlate errors with browsers.
That can be quite limiting, especially if you're trying to support MSIE as it has a hard cutoff at ~1000 characters for the URL (so you're limiting yourself to below 1k of data, which may not even be enough for a slightly deep stack trace).
