Playing movies off our iPad at the cottage
Your best, but not least expensive, option is to buy an Apple TV.  

This will allow you to AirPlay movies from your iPad (as well as any other Apple device) to your HDTV without using any cables.

I personally use my Apple TV to do this exact thing on TVs where we don't have a cable connection.

You can buy a new or used one but I wouldn't buy a used one older than version 3.

The movies you're trying to play using the cable might have HDCP copyright protection.