Old laptop power brick > DIY current regulator > 20w COB LED - will it do what I think it will?
You've said the LED lamp is \$20\:\text{W}\$ and I'll assume that's because it's written down, somewhere. Your measurements are consistent with the rating, too. So I'm going to accept the numbers you developed for the LED lamp.

Your laptop charger system is another issue. Modern ones are sophisticated enough that they may require some kind of communication in order to active their power supply activity. If so, you may have some difficulties getting it to drive your LED lamp. But let's assume that's not a problem, here.

Given the voltage rating of your power supply and its compliance current, it would seem to be a decent fit. Probably the easiest thing to do is to buy a few \$10\:\text{W}\$ ceramic resistors, with values from say \$3.3\:\Omega\$ down to \$1.8\:\Omega\$. Start with the highest valued resistor and place it in series with the LED lamp and your power supply (with correct polarity!) Measure the voltage across the \$3.3\:\Omega\$ resistor. Divide the voltage you measured by this resistance to work out the current through the resistor (and therefore through the LED lamp.) If this current is at or below the figure of \$1.2\:\text{A}\$ (or so close it doesn't matter much), then you have worked out a match-up. If the current is too low, then replace that resistor with the next lower value and repeat until you get the right current.

You could go to the trouble of adding a pair of BJTs to provide a current limiter. But there probably isn't any point. You still have to dissipate some excess power either way and the main advantage for the BJTs is supporting a wider range of source voltages -- something you don't care to do. So it's probably pointless worrying about anything more than a simple resistor to limit the current with your existing power supply.



Just in case you might have gotten some numbers wrong, it might be a good idea to also include at least an \$18\:\Omega\$ and perhaps even a \$33\:\Omega\$ resistor in your package. (\$10\:\text{W}\$ also.) Put one of them in series with your LED lamp first, just to make sure. Switch it on for a few moments and check the voltage across it, quickly. Let the resistor warm up for a few seconds, too. You should be able to verify that the current is fairly low by making the calculation above and also should be able to verify that the dissipation in the resistor is modest -- it doesn't heat up too fast.



If for any reason any of the resistors appear to heat up very quickly (which may happen if the LED lamp isn't dropping much voltage), then turn things off immediately and do NOT let the resistor get too hot. In this case, it would appear that your LED lamp needs a much larger voltage drop across a resistor and this will require you reporting more detailed results from these experiments before a better answer can be offered.