Technically, DVI is HDMI but without the audio channels, so you could get an adapter or a DVI to HDMI cable to hook up your current system to a TV without audio.
Anyhow, the PC you have still uses the normal ATX sized power supply so lets just find a replacement PSU instead of buying a worthless piece of crap supplement PSU for 3x more expensive. Before we search, lets see how much power your system nets.
Intel i5 2400: 95w tdp
Intel H61 chipset: 6.1w tdp
GTX 560 ti: 170w tdp
1TB 7200 hdd: 10w tdp
So at max load, you system will chew through around 300w, give or take. Add a little PSU overhead of about 20% and are around the 350-380 mark.
I'd look at the Corsair Builder Series CX430 V2
PSU for $25AR + $2sh right now on newegg. It's cheap, great, more than enough power to match your system and its JonnyGuru recommended. Alternatively, you can look at the bigger brother, the CX500 V2
PSU for $30AR, AC (EMCNJJN29) that is just way more than you'll ever need.
Also probably nothing to note, but I would check the graphic card's clearance and make sure it can fit the 9" card. Judging by the photos, it does look like it should have room, but still; better to make sure than having to pay a 15% restock fee for not. It actually looks pretty room for a compact HP pre-built system.