So a little late to the party but I figured I'd throw my two cents in since I have a similar setup. Looking in to it all drove me nuts for a bit so I'll get a bit wordy here to save you some time.
Like americanaudiophile said, there is a chance that your tv wont be able to output to it's own internal speakers and through a source like RCA out at the same time. Some do, some don't, so you really want to nail that down before buying any gear. If you have an optical out, that typically cuts off the internal speakers for sure, but it provides a very clean audio source and is great for doing headphone/soundbar setups.
So a couple different options for under $500
First, unless you already have a good pair of headphones (I'm assuming you don't), make life easy and ditch the long cables and amp. You can get a decent set of wireless RF headphones for $150-$250 easy enough. Avoid Bluetooth, unless you're getting a low latency setup the audio won't sync with the video of what you're watching. Real quick terminology for looking at headsets, open backed headphones leak sound out, closed back won't leak sound (or at least as much sound).
Sennhiser makes a good couple of models that may be in your budget:
Sennheiser RS 165 RF $167. Rechargeable from the base station, over ear, RF signal. Audio input from a 3.5mm jack.
Sennheiser RS 175 RF $215. Same as above, but does optical or 3.5mm jack for inputs
Now, the rest depends entirely on what your TV can do.
If you can output through the RCA outputs and through the internal speakers at the same time:
Fantastic, your setup is dead simple. Buy the headphones, buy a RCA to 3.5mm cable ($8) or RCA to 3.5mm adapter ($6) off amazon or your vendor of choice, plug them in and you're good to go. You'll get audio out both sources and can control the headphones independently. The headphones I linked have audio controls on the side of the headphones themselves.
If your TV cuts off the audio to the internal speaker when using the RCA out:
Your best option (though pricier) is to get a sound bar like americanaudiophile suggested. There are a ton of options out there, and honestly they usually sound much better than the built in speakers. Just make sure you find one that take RCA inputs. I personally like Samsung models, but Sony, Vizio, and many others will make ones in your budget. From there you split the audio coming out of the TV with a RCA splitter (something like https
/www.amazon.com/Bonayuanda-Female-Splitter-Adapter-12inch/dp/B01J007XGO/ref=sr_1_5?s=electronics&ie=UTF8&qid=1522970326&sr=1-5&keywords=rca+splitter) and then plug your devices in. Again, as with the above suggestion, you'll need an RCA to 3.5mm adapter for the headphones.
If your TV only has an optical out:
This the most common output on most TV these days. Buy a sound bar like suggested above making sure to pick one with an optical input, two extra optical cables, an optical splitter (this is the one I use https
/www.amazon.com/gp/product/B00G191YL8/ref=oh_aui_search_detailpage?ie=UTF8&psc=1) and plug in your devices. With the headphones listed, if you were to go with the RS 165s above, you'd need a optical to 3.5mm (https
/www.amazon.com/PROZOR-Digital-Converter-Toslink-Adapter/dp/B00KNNSKV0/ref=sr_1_1?s=electronics&ie=UTF8&qid=1522970422&sr=1-1&keywords=optical+to+3.5mm). The RS 175 have optical built in so they'd be an easier fit in to the system.
Thats a lot of info I threw at you, but the basics of it all come down to splitting the audio source and putting it in to headphones. Going with wireless headphoens will free up any long cable issues and allow your grandmother to sit anywhere she pleases or getup and move around without being tethered to a spot. Something I'm sure she'd enjoy.
In terms of cost, $167 for headphones + say $200 for a decent quality soundbar, + $20-$40 will run you about $400 bucks and keep all the equipment out from underfoot and give a lot of flexibility. If you're not super picky about the sound bar, then closr to the $300 mark. If your grandmother has uneven hearing loss, you could splurge on something like the RS185, which gives the ability to set different audio balance on the headphones (also does straight RCA input as well as optical) but they run $300+.