SQL Server 2008, cpu 100% until recreate an index
My gut guess is that the CPU usage dropping after recreating the PK is coincidental, especially since you believe the index is not fragmented. My guess is there's nothing wrong with the index, per se. More likely, it's the lack of indexes to satisfy the particular queries / procs executing against that table.

What you need to do is monitor the server. This is 2008 so you have many tools available to you and you can go about this a couple different ways.

It sounds like this is happening frequently throughout the day, so observing a Profiler trace and sorting on CPU can be the quickest way to identify what queries / procs are hammering CPU at that particular moment. (SQL Trace is the server side version of the Profiler GUI - this might be a better option if you need to trace for an extended period of time, but that is a different conversation.)

Along the same lines and a bit more accurate would be to correlate a Perfmon trace with a Profiler trace and then you can see exactly what was running when your CPU spiked. (You can google that up for instructions.)

Another approach would be to look at the execution DMV's (see link below) and again sort descending on total_worker_time as that relates to CPU cost.

These approaches will narrow down the queries / procs you need to focus on and from there you need to look at the excecution plan and determine if you can add indexes, break up the query into smaller pieces, etc...

Here are some links to get you started:

https://www.sqlskills.com/blogs/glenn/category/dmv-queries/

http://whoisactive.com/