Several leds 5v and 12v. Limited space.

I have several leds to power and limited space.

I have
1x 5v RGB LED string that requires around 4Amps 67LEDs

1x 5v RGB LED strip with 180 LEDs which I think requires around 9Amps

And
16x 12V 3 watt G4 LED bulbs

What's the most compact way of powering these lights? Is it better to have a 240v ac to 12 v dc supply then step down to 5 volts? Is there an IC that I can use to make the arduino control and dim 16 12V leds?
For reference this is the sort of space I'm dealing with. No space for anything bigger than 8cmx8cm though there is space for several thin transformers

RE: Is there an IC that I can use to make the arduino control and dim 16 12V leds?

Yes.
Use PWM from the Arduino and control a mosfet.
Or search for "dimmable LED controller IC"

RE: Power Supplies

You have, at least, 4 options:
Use Google to locate:

Option #1: A Single 12V P/S

Input: 240 VAC
12V DC Output at 11 amps (minimum)
You build a 12V-to-5V Buck Inverter

Option #2 A Multi-Output P/S

Input: 240 VAC
5V DC Output at 13 amps (minimum)
12V DC Output at 4 amps (minimum)

Option #3: A 5V P/S and a 12V P/S

Input: 240 VAC
5V DC Output at 13 amps (minimum)

Input: 240 VAC
12V DC Output at 4 amps (minimum)

Option #4: Two 5V P/S and a 12V P/S

Input: 240 VAC
5V DC Output at 4 amps (minimum)

Input: 240 VAC
5V DC Output at 9 amps (minimum)

Input: 240 VAC
12V DC Output at 4 amps (minimum)

Is this even possible in 8x8 cm ?

Led lights use a adapter/inverter/thingy that supplies more or less a stable current for the leds. They don't need to be accurate and can be very small.

You want a stable 12V and 5V and many amps.
I think mrsummitville mentioned everything. Try to find a single power supply with 12V and 5V output (Option #2) that is small enough.