The answer is: "Yes" and "No".
It is indeed possible to modify an RGB LED kit to accommodate a longer length of RGB strip. However, there are two types of limitations:
(a) The limitation of the power supply lying inside of the existing RGB driver unit (Fig 1)
(b) The limitation of the RGB LED strip itself when connected as a single stretch of strip (Fig 2), rather than cut into shorter strip segments which are branched at the source (Fig 3)
In this instructable, I will be teaching you how to overcome the limitation in (a), which surprisingly, turns out to be an easy thing to do.
Note that a better approach would be to buy the RGB LED controller and the power supply as separate units. However, there are cases when you'd still want to apply this instructable.
So let's begin!
[The figures in this instructable are created using Inkscape, which are provided in Inkscape's native format for unlimited and unrestricted use!]
Step 1: Buy a Separate Power Supply to Replace the RGB Kit's Internal Power Supply
Any 12VDC power supply with 10 amps or more would do the job, and would allow you to drive double the original length of the RGB LED strip without flickering, heating up, or generating plenty of switching noise (all of which are signs of malfunction associated with an underpowered internal power supply).
Note that it is obligatory to split the length of the RGB strip into 2 or more shorter segments, and branch them at the root as demonstrated in Fig 3 in the introduction, otherwise, the system would work initially but would not be reliable due to the limitation of the RGB strip itself. (Fig 2 in the introduction)
Step 2: Take the PCB Out and Examine It
Needless to say, the kit's adapter should be unplugged from the AC as you take it apart. Taking it apart is as easy as pressing the side clips with a flat screw driver and cracking it open (a non-destructive operation).
Once opened, the PCB is taken out and flipped upside down, you should be able to see that the outlet's AC voltage is fed from one side, while the other low-voltage inputs/outputs are placed at the other side. In fact, there is often a transformer in the middle that separates the circuit into two halves/section which are electrically isolated: the AC (high-voltage) half and the low-voltage half. (Check the second photo which is attached to this step)
Step 3: Identify the (+) and the (-) Poles of the Supply Voltage in the Low-voltage Half of the Circuit
Sounds complicated, but is actually quite straightforward. In the low-voltage half of the circuit, a supply voltage of 12VDC is created and maintained using an "output capacitor" (capacitors are components that store electricity), which is rated at a lower voltage than the input capacitor in the high-voltage half of the circuit.
Simply, by looking into the lanes into which the (+) and the (-) leads of the capacitor connect to on the bottom of the PCB, we are able to discover the right points of tapping into our system, without actually needing to run the system and test it while running (which would be a hazardous operation unless you are well-versed in electronic testing).
Having identified the (+) and the (-) lanes on the bottom of the PCB, solder two reasonably gauged wires into them, and let these two wires replace the AC input wires, which can now be safely removed (desoldered or cut) from the PCB.
Step 4: What Did We Actually Do?
The idea behind this modification is to simply "bypass" the internal power supply of the RGB kit's driver unit or adapter, which works pretty well and does not require further understanding of how the system works. In fact, after the modification, the "high-voltage" half of the circuit will be permanently unused, leading to any symptoms associated with a poor internal power supply to disappear from the system immediately (such as overheating and switching noise).
The first time you run the system after the modification, expect it to take a second or two to run, but after that, it would run indefinitely without problems. Note that a possible enhancement of this modification would be to remove the capacitor from the low-voltage half of the circuit, but it is not required.
Thank you for reaching this far, and feel free to leave your comments, feedback and personal experience.