Possible to tell what a query was in session I just killed?
By default nothing keep tracks of all running query in SQL Server.  Depending on what the query was you might explore few things. None of these guarantee that you will find what you are looking for.


sys.dm_exec_query_stats (Transact-SQL)

Returns aggregate performance statistics for cached query plans in SQL Server. The view contains one row per query statement within the cached plan, and the lifetime of the rows are tied to the plan itself. When a plan is removed from the cache, the corresponding rows are eliminated from this view.

You will need to join these with other DMVs. Link has examples.
A Few Cool Things You Can Identify Using the Default Trace by Greg Larsen
Use the system_health Session if you still have the trace files.
Since you mentioned SQL Server not responding I would also look at SQL Error log, Windows event logs.