Sort Old Data into a Pivot Table

I am currently working with a very old spreadsheet that is full of data and was poorly designed. Criteria is displayed both vertically and horizontally, with the data falling in a 'matrix-style' within. (I have attached a mock spreadsheet that illustrates this). I am looking to use formulas in order to sort the data into columns so I can display it in a pivot table.

I want to sort the information in the Data table so it totals the number of contracts per product per month.

I have tried using an SUMIF to sort the data into columns. This works, but the formulas are not very easy to pull down, and it isn't very user-friendly for the others who will be using the spreadsheet.

The attached file explains things a lot better than I can in writing.

I appreciate any help you can provide. Thanks a lot.

Pivot Table From Raw Data.xlsx

I am still having trouble with this and would appreciate any help I could get. Thanks.