I wouldn't use a 5V PSU unless I had a really nice 5V fan in stock. 12V fans are much more readily available and far easier to control the speed of - if you get the 4 wire PC variety, as your speed control circuit only has to provide a PWM logic signal, not chop the full fan motor current. However, assuming you have a 5V fan handy and intend to use that PSU board, before you do so, lets see the underside of the PCB so we can check clearances etc.
As you may have gathered, I'm not a great fan of using wallwarts inside equipment,unless you can fit a chassis mount mains socket on a bracket to plug them into, and secure them to the bracket with a couple of cable ties so they cant work their way out of the socket, and also keep their ambient temperature under 40ºC, so they aren't stressed more than they'd be on a hot day with no air conditioning.
BODGE AT YOUR OWN RISK!
However, if I was bodging it for home use, I'd try to keep the base of the wallwart case, removing the prongs and drilling it for some Nylon bolts to fasten it down. Then, if there's no good place to drill holes to secure the PCB, I'd secure the PCB in the base with a bit of neutral cure (electrical grade) silicone caulk at the edges, taking care not to bridge tracks on the primary side or from primary to secondary. I'd also put a cable tie right round the base and board, over the transformer with a scrap of closed cell foam to pad it where its pushing on the winding, so its still firmly held if the caulk lets go. The mains wiring would be fastened into the base with small cable ties (and extra holes drilled for them in the base) and lead out via two holes drilled the opposite end to the notch for the 5V output cable. The output cable would similarly be cable tied to the base.
If you can manage to thermally insulate the heating element from the base electronics chamber, (preferably without using polystyrene in contact with any electrical part or wiring - it interacts badly with the plasticisers in PVC wire insulation, especially at elevated temperatures) consider adding a second fan to draw air through the base to keep the electronics cool.