Queries are taking long time to execute
Your PROCESSLIST

Look back the processlist in your post. I see you are using INSERT DELAYED. I have always felt it was invented as a band-aid to squeeze some write performance out of MyISAM. Note that the first line of the INSERT DELAYED Documentation says


  The DELAYED option for the INSERT statement is a MySQL extension to standard SQL that can be used for certain kinds of tables (such as MyISAM). When a client uses INSERT DELAYED, it gets an okay from the server at once, and the row is queued to be inserted when the table is not in use by any other thread.


For those using MySQL 5.6, INSERT DELAYED IS DEPRECATED:


  As of MySQL 5.6.6, INSERT DELAYED is deprecated, and will be removed in a future release. Use INSERT (without DELAYED) instead.


You should consider using INSERT instead of INSERT DELAYED. Why? From the Documentation:


  INSERT DELAYED is slower than a normal INSERT if the table is not otherwise in use. There is also the additional overhead for the server to handle a separate thread for each table for which there are delayed rows. This means that you should use INSERT DELAYED only when you are really sure that you need it.


What is the permanent and long term solution?

You need to get away from MyISAM. Any kind of writing you do against a MyISAM table will always require a full table lock. Don't get me wrong. There are rare occasions when MyISAM is better than, and could even outperform, InnoDB :


Sep 20, 2011 :Best of MyISAM and InnoDB
May 03, 2012 : Which is faster, InnoDB or MyISAM?


At times, MyISAM could also have a slight edge in read performance over InnoDB. Why? Disk I/O wise, MyISAM has a slight edge with ROW_FORMAT=Fixed because you only interact with one file, the .MYD file. The row size is completely predictable because VARCHAR fields are treated as CHAR. This will reduce access time for data retrieval.


May 10, 2011 : What is the performance impact of using CHAR vs VARCHAR on a fixed-size field?
Mar 25, 2011 : Performance implications of MySQL VARCHAR sizes


In the short term, just ditch INSERT DELAYED. It seems that your write load has increased in overhead beyond what MyISAM can handle.

After doing that, give it some time. If you still see perform issues, it may be time to switch all MyISAM tables to InnoDB (See my old post When to switch from MyISAM to InnoDB?)