I was finally able to fix the repeat header problem. The suggestion from Doug Robbins led me to the answer.
I was checking in the Table Properties to see if Text Wrapping = None was enabled (it was). I then noticed that I could look at Cell Properties as well. With only the header row highlighted, I checked the Cell Properties and then clicked on the Options button. I noticed that the option for Wrap Text was checked. I disabled it and saved the change.
I then went back to the Row Properties and rechecked the option for "Repeat as header row at the top of each page". I saved the change and that was it. The header appeared at the top of each page!
I hope this helps others with the same problem. Thanks to Doug and to Rich for their help.