Luckily for you the genius programmers at InterWorks have made a wonderful tool that does this.
We offer a free 14 day trial and you will be using just a piece of this tool called Data Source Audit. It will let you see per data source how many fields you have and it will tell you if they are a dimension or a measure and even the data type (date, datetime, int, float, etc). Once you install the product use the Data Source Audit on your workbook that you have in question and then you will be able to export the data into Tableau, Excel, or a PDF. The export will have this information that you are after. It's quick and easy to get the total number of Dimensions, Measures by datasource with this fantastic piece of software.