Optionally: Throw away/recycle away the supplied chinese noname power supply. Buy a used laptop PSU from a reputable brand locally for cheap instead. I scored a Lenovo 135W/20V laptop PSU for $5 at my local Goodwill equivalent. Solder on a 5.5mm barrel jack connector.
1. His speakers are powered already. He doesn't need an amp.
2. Even if they weren't, how is he supposed to connect to the Fosi without a headphone jack coming out from the TV? The Fosi only has RCA input.
You suggested an amp. The fact that I'm able to use a headphone jack to connect my speakers should tell you I don't need an amplifier.
The question posed is how to connect those speakers if I no longer had access to a headphone jack. Currently the headphone jack is working fine.
For what it's worth, here's a comment that seems like it's get a perfect solution for me.
Find a tiny TPA3255- or TPA3116-based amp. These are class D amplifier chips made by TI.
https://en.wikipedia.org/wiki/Class-D_amplifier
Buy one of these from e.g. Amazon.
Optionally: Throw away/recycle away the supplied chinese noname power supply. Buy a used laptop PSU from a reputable brand locally for cheap instead. I scored a Lenovo 135W/20V laptop PSU for $5 at my local Goodwill equivalent. Solder on a 5.5mm barrel jack connector.
My fav for your use case: Fosi Audio TB10D.