If you can find the model # on the back of the tv we can look up the specs or manual to see what it has for connectors for DVDs, VCRs, cable boxes, etc. If you know that the DVD works you could set your tv to play a DVD, and then at that point, turn off both the tv and DVD, disconnect the DVD at the back of the tv and plug the Nextbox into that set of connections. With those type of cables and connectors you have to make sure that they are pushed all the way in, otherwise there is no valid connection for the video or audio. Turn on both the tv and Nextbox and you should have video and audio running.
There are switches available for those type of connectors which allow you to plug more than one item into the switch, such as stand alone DVDs, VCRs, etc, and then you would use the switch to decide what you wanted to watch on the tv. From the switch the connection to the tv is a single cable set, like the one that you already have, which is a video cable combined with left and right audio cables. So, that is an option as well, or as Gdkitty suggested, go down to the lower model and use the antenna or cable input on the tv, set for channel 3 (?) to watch the Nextbox.