Your actual question is
If LEDs are suppose to be super energy-saving, why are they giving off so much heat
to which the simple answer is
"The very best LEDs that modern physics can make well enough for them to be able to be commercially used will dissipate 80% or so of their drive energy as heat.
LEDs used in a cheap light can reasonably be expected to turn 90% to 95% of their drive energy into heat. "
As I noted recently - the BEST LEDs are around 20% efficient in converting electrical energy to light. 20% is extremely good for an LED - just not extremely good in absolute terms. Most are less efficient.
It was attached to a heat-sink that I removed and discarded, thinking nothing of it.
The ability of a heatsink to work depends on it being in thermal contact with the heat. Alas, no matter how little you think of them they are still needed just as much if they were required in the first place. Removing them stops them form working as intended in all known cases to date.
I recently took apart a cheap ($7) LED light to use in a project
A $7 light probably cost under $1.50 at a factory door somewhere in China. Makers of products that cost that little tend to include nothing that is not utterly essential. If it is there it will probably not work well if you remove it, whatever it is.
11 tiny LED
If the LEDs are white they are probably rated at 20 mA each and drop about 3V.
At full rated output power in = V x I = 3V x 20 mA x 11 LEDs = 660 mW. A circular piece of metal large enough to accomodate 11 x LEDs would probably get hot but not very hot i still air so the LEDs are probably being run at above rated current - this is very usual and would cause them to die rather quickly in normal use. They will die much much much more quickly without a heatsink.
In some systems of units, luminous flux has the same units as radiant flux. The luminous efficacy of radiation is then dimensionless. In this case, it is often instead called the luminous efficiency, and may be expressed as a percentage.
A common choice is to choose units such that the maximum possible efficacy, 683 lm/W, corresponds to an efficiency of 100%.
The distinction between efficacy and efficiency is not always carefully maintained in published sources, so it is not uncommon to see "efficiencies" expressed in lumens per watt, or "efficacies" expressed as a percentage.
The luminous coefficient is luminous efficiency expressed as a value between zero and one, with one corresponding to an efficacy of 683 lm/W.
On that basis, the 20% efficient figure that I mentioned would be
683 x 20% = 137 lumen/Watt.
I have used large quantities of the utterly marvellous but relatively low power (50 mA max) Nichia NSPWR70CSS-K1 "Raijin" LEDs which have been on the market for about 3 years. These were well ahead of any other commercial offering when they first came out and even now are extremely good.
At "full power" of 50 mA they are rated at slightly over 120 l/W.
At 30 mA they are typically specified at 165 l/W (depending on binning).
165/683 ~= 24%.
At this point they are outputting about 20 mW of actual light energy = 20 x that of a Class 1 LASER diode. The light is however vastly more diffuse (80 degree radiation angle).
According to tests which (to my surprise) Nichia were kind enough to do for me, they actually constitute a formal minor eye hazard at the blue end of the spectrum, but you would have to be beyond perverse to look into the LED from close enough for long enough to acquire eye damage from a 80 degree 150 mW white LED.
The 24% efficiency is exceptionally high even by current standards - but whereas the Raijin is rated at only 90 mW input at 165 l/w and 150 mW in at 120 l/W, current 3W to 6W LEDs are quoting l/W figures in the 120-140 l/W range at significant percentages of their full power and over 170 l/W at low % of full power in some cases.
Such results are attainable only from top bins of top performance parts and my suggested 20% efficient = 136 l/W upper guideline is still in excess of what you would expect from the large majority of devices currently available. No name parts from Asia (as are liable to be used in the light in question) will almost always be well below this level.
Note that the lumen is defined with the human eye spectral response as part of the specification. Thus red LEDs are easier to produce with high l/W ratings than are white LEDs and deep blue and near UV LEDs have such low l/W ratings that they are specified instead in mW output.
The chart below (said to be from the U.S. Department of Energy) which shows a lumen per watt comparison of common light sources and white LEDs as well as the projection for the future.purports to show past and future efficincies of various light sources.
I have added a few real world samples and comments.
Note their "White LED lamp" line, which is about correct so far. These are handicapped by needing to operate from AC mains and provide power factor correction and in many cases dimmer compatibility. Energy conversion efficiency from mains to LED input is often under 80% and they usually use diffusers to 'improve' user perception of the light source so, if a diffuser reduces output by 10%, a delivered 80 l/w means the LEd will be 80/80%/90% =~ 110 l/W.