Reddit mentions of iMBAPrice 5V DC Wall Power Adapter UL Listed Power Supply (5-Feet, 5V 2A(2000mA))

There are really only four things you have to watch for when choosing a power adapter: connector size, voltage, current, and polarity.
Input voltage is what the device in question needs to function; your headphone transmitter for example. Output voltage refers to a device, often a power supply, that delivers or sends out a certain amount of power; the thing you don't have for example.

For polarity, the same rule will almost always apply in the same way for all electronics. Yours is no exception since the pictured symbol indicates that the center pin is positive (+) and outer case is negative (-).

For voltage, 4.5v power adapters are fairly uncommon in my experience. Since your device is battery powered, it will almost certainly be able to accept an extra 0.5v on its input. 5v power adapters are also a lot cheaper and more readily available. You might even already own one.

For current, all you ever have to check is if your power adapter has an equal or greater current capacity (mA, A, milliamps, or Amps) then what the device requires. since your device is a simple headphone/audio transmitter, I'm going to guess it dosn't require much current. Maybe 500mA at the most. With that being said, 5v 2000mA power adapters are still as cheap or cheaper then most of the lesser powerful ones.

I'm going to take a wild guess and say that the jack pictured is a 2.1mm (pin) by 5.5mm (outer sleeve). This is among the most common sizes and seems to scale to your picture nicely.

In conclusion, here is what I would get:
5V DC Wall Power Adapter ($7.29 + Free Shipping)
If on a budget, this should do just as well (though might take couple weeks to ship):
Generic USB Barrel Jack Cable ($1.71 + Free Shipping)

Just got mine, they were also out of stock. I went with one of these:
https://www.amazon.com/gp/aw/d/B00GUO5WUI

As always, it's up to you to triple check. 2.1x5.5mm, center positive, 5v, at least 1amp. That link seems to check all the boxes, and is UL listed for what it's worth. I've used it for a few hours and it seems OK. Agree do not use higher VOLTAGE though higher AMPERAGE is ok. Their site still says they're not responsible for damage due to power supply that isn't theirs.

Good luck!

What he is saying is to get a power adapter like you would use with your router or external hard drive. That plugs into your wall and then outputs DC voltage. This would negate the need to use this AC/DC converter.
Such as this
https://smile.amazon.com/iMBAPrice-Adapter-Listed-Supply-5-Feet/dp/B00GUO5WUI/ref=sr_1_4?keywords=5vdc+adapter&qid=1574732473&sr=8-4

Then you could use this to power your circuit instead of what you have.

And to go a little further. If you want to use what you got. Which is fine. After soldering your wires for the AC. the L is line. Which is usually black and is the HOT wire. The N is for Neutral which is white in US. IF you do that make sure to use some heat shrink or rtv silicone as an insulator so you or someone else doesn't get shocked. Then you can use your converter to power your blinky board. Don't run the AC power through your prototype circuit.

Also remember this is electricity. Do a few youtube videos on arc flash and realize what you are messing with can kill ya. Now you not gonna arc flash anything with that circuit, but 120VAC has the tendency to contract your muscles your holding it with and then you can't let go.

Don't be a dumbass if you wanna go back to high school after your break. And keep asking questions.

I dunno, 2.5x0.8mm is pretty easy to find on amazon. The same goes for the 5.5x2.1mm, that one is probably the easiest as a number of them are made with plugs that work in 5.5x2.2mm as well.

When I was doing research for a similar type of strip light (color changing neopixels) 12 V was definitely overkill. I used 5V 2A similar to the one I linked. Worked well for my project.

https://www.amazon.com/dp/B00GUO5WUI/ref=cm_sw_r_cp_api_i_cb46Cb0GVBYG0