After extensive testing, I've found a solution here. The tables must be "complete" which by ENML standards means they have to have a header, body, and footer--you can't just go right into rows and cells like a classic HTML table. See below for a template table that you can use as a starting point.
<table> <thead> <tr> <th>Col1</th> <th>Col2</th> </tr> </thead> <tbody> <tr> <td>R1C1</td> <td>R1C2</td> </tr> <tr> <td>R2C1</td> <td>R2C2</td> </tr> </tbody> <tfoot> <tr> <td>R3C1</td> <td>R3C2</td> </tr> </tfoot></table>