It isn't clear if your older TV has a DVI or DisplayPort connection. So, you'll need either an HDMI to DisplayPort or an HDMI to DVI adapter.
If those assumptions are correct, you're now looking at weighing the cost of things.
Roku Streaming Stick+ is $50 (though you can often find them on sale).
HDMI (female) to DisplayPort adapters run from $15-40.
HDMI (female) to DVI adapters are around $30+ and you still need DVI and audio cables.
Where all this is leading is to consider the cost to do what you're seeking vs the cost of a similar size Roku TV. Of course, cost isn't always the most important factor, but it may be worth considering.
DBDukes Roku Community Streaming Expert Note: I am not a Roku employee.
If this post solves your problem please help others find this answer and click "Accept as Solution."