Are we a good fit for SQL Server Enterprise and table partitioning?
My question is, do you think we should look into upgrading to SQL
  Server 2014 Enterprise so that we can partition our time sheet tables?


No. Absolutely not. On a pocket change server of 8 cores you would be spending ~$50k and be unlikely to see any benefit.

I would suggest trying, in this order:


Increase the memory allocation to SQL Server. You said the server is dedicated but have only half the RAM allocated. 32GB server, allocate 28GB.
Upgrade the server memory to [size of database] + 4GB.
Post the problem queries on dba.se.
Engage an expert.