Hi, to make a joke of it: I loathe power supplies - they look so simple but in fact have a lot of important details to be certain of first!
Is that 2A AC or 2A DC? ...5A AC or 5A DC?
You may need to re-size a few components to go from 2A to 5A output, but not necessarily. Also, maybe the 5A transformer will be somewhat larger and heavier than the current 2A one.
If you can find the diode datasheet, look for the peak current capacity, as there's inrush current when the supply is turned on.
May need a bigger capacitor if that blue one is only 1000uF, and may want to bear in mind the ripple current the capacitor can handle.
Does that thing have a fuse on the primary input and one on the secondary output side? Ideally, yes.
If that diode is a 1N4148 type, what is it for? They are relatively low current, so i guess it is out of/not in series with the 2A output line.
Whatever the HHK black box thing is, you need to read the datasheet for that too.
The transistor should be fine. No idea about the heat-sink 'though, sorry, you would think that it wasn't designed expecting 5A across the transistor but only for the minimum dissipation possible for mass production cost-saving.
Derate as per instructions provided by manufacturers, maybe 6A diodes at 5A is fine but not the ideal.
Use a fan to dissipate heat if you need one, and can fit it into the case.
...to give a short answer, personally I think it would be a case of having to replace so many parts (e.g. diodes need to be rated for at least double the current they will pass; worse still, the case/enclosure to fit the new transformer, if it's larger) - possibly also down to the resistor for the LED - that it would perhaps be (quite a lot) cheaper to buy a 5A/15V battery charger.