We ended up with a corupt msdb after a power outage. Restored the msdb backup, and noticed that emails were not going out after jobs completed. Went into database mail, got the "must enable service broker" message, and experienced the hangup described above. Simply stopping SQL Server Agent before going back into setup Mail fixed the problem.
Why wouldn't Service Broker be enabled in the backed up copy of msdb? All was working for months before, and a the backup was a few hours old.
Student of SQL and Golf, Master of Neither