^ Finding exactly the right size replacement transformer that will solder onto the space on the PCB could take a very long time and you may never find it, and it may be very expensive including shipping, compared to the low value of the speaker amp.
12VDC output AC/DC wall adapters on the other hand, are very common and inexpensive, and typically it is easy to find some rated for 1A which is probably higher current rating than that tiny original transformer was capable of, which is why I suspect it overheated and failed in the first place.
There is no reasonable/safe way to convert 220VAC to 12VDC without a transformer. This is a pretty simple repair, why are you wasting time questioning people who realize how to fix this with minimal time and expense, since it is about the cheapest, crudest low end speaker amp possible? It really is not worth great time or expense to fix this.
As Minder mentioned, you should solder the wires from an AC-DC 12V adapter to the PCB. The positive lead wire from the adapter would be soldered to the hole labeled "P9" on the PCB and the negative lead to the hole labeled "P10". If you prefer you could keep the barrel plug on the AC/DC adapter and buy a barrel socket to mount on the speaker cabinet, then solder wiring from P9 and P10 to the respective terminals on the barrel socket, but I question whether this amp is worth buying any more parts for than absolutely necessary.
Note that you don't necessarily even need to use an AC/DC adapter. If it is a desktop computer so it has a source of 12VDC, you could just power this amp board from that, just get a $1 connector for the PSU 4 pin molex plug and solder that to the amp P9 and P10, maybe needing to add a piece of wire to extend the length.