Not really my area of expertise.
But I bet I'm more qualified than most people.
(Mechanic, machinist, fabricator, welder, computer geek, etc)
Looking at the Dragonhawk Tattoo Gun shown in this example,http://www.amazon.com/DragonHawk-Tattoo-Color-Deluxe-Case/dp/B0055EIFC6
,I see the gun works with Solenoids.
The coil is energized with electricity, and pulls the iron cylinder core back, and forth.
This cylinder then hits a flat metal bar, and the bar is connected to the tattoo needle. (Through the various mechanism)
A) Your power adapter is an AC adapter. It converts the AC voltages from your home, or business, into low DC Voltage.
First thing to check is that it IS delivering power.
LED light lit up on the AC adapter means it is GETTING power, but we do not know if it is delivering power to the gun.
A multimeter is used to check.
Now wait a minute...............
An inexpensive multimeter can be purchased for as little as $5 to $12, if you do not have access to one.
Available in a multitude of stores.
An auto parts store is but one example.
I have seen them on checkout aisle racks, at major discount stores.
Looking at the AC adapter, what style of plug does it have, for connecting to the gun?
I need to find out what the Positive connection is, and the Negative connection.
The multimeter Function Knob is set to DC Voltage.
If just a symbol, the symbol is a dotted line over a solid line.
[IF there is more than one DC Voltage scale, set it to the 0 - 50 Volts scale.
0 to 50 Volts DC ]
The Positive (Red) probe lead of the multimeter, touches the Positive connection, of the plug that connects into the gun.
The Negative (Black) probe lead touches the Negative connection, of the plug that plugs into the gun.
Voltage is read. Don't know what the voltage is, but YOU will.
Look at the AC adapter's label.
Input: 100 ~ 240 Volts / AC 50 ~ 60Hz
We need -> Output: xxVDC @ xxA
(xxVDC is how many DC Volts.
Let's use 12 as an example.
xxA is how much Amperage.
Let's use 3.67A as an example
3.67Amp's, or 3.67 Amperes. Same/same)
AC adapter is putting out the required voltage?
Then the switch is checked next.
Check switch for continuity.
[ Check with multimeter set to OHM's. 1K.
NO AC adapter connected to gun.
Positive ( Red) probe lead to one side of switch.
Negative (Black) probe lead to the other side of the switch.
Turn the switch to on.
If the switch is good you will get a reading on the multimeter.
DOES NOT matter at this point what the reading is.
Just matters that you DO get a reading ]
Switch is good?
Then there is a problem with one of those coils, going around the cylindrical metal bar. Perhaps a wire going to them is broken off.
Or wires in the coil have burnt.
There is a set screw for adjusting how far the vibrating metal bar, is away from the cylindrical metal bar.
Determines the distance of the flat vibrating metal bar, to the cylindrical metal core in the coil.
If this has been adjusted too tight, the flat metal bar won't vibrate.
Doubt this is the problem, or you would have stated the gun 'Hums', but won't vibrate.
[The black cylindrical things are coils. The shiny cylindrical metal objects inside them are the vibrating cores ]
I suspect the AC adapter isn't putting out any power, or the switch on the gun is bad.
Post back in a Comment, about the design of the plug from AC adapter, that plugs into the gun.