The link that you supplied for the LED specifies that it need 300 mA, which is too much for the Arduino (for each GPIO, the maximum is 40 mA, for a total of all GPIO not exceeding 200 mA, unless you have a special board...). Try using an independent power source as source for the regulator. If green or blue LED, you need more than 3V. The link you supplied indicates that for Red, you have more voltage than required (2.4 to 2.6V). With 5V at the grid, with 0 V at the source pin, this NMOS could handle a max of 70 A from its drain to its source pin, so this is surely not a problem here. Since there is almost no current ( at very low frequency) through the grid pin, adding a resistor there won't do much. The Arduino may also have a problem to sink 300 mA (it should handle 400 mA, in total, so who knows... ), so plugging the voltage regulator to an independent power source could also help there too. If the LED does not turn off as soon as the Arduino logical output asks it, you may have to add a pull down resistance to the gate, though, but that resistance won't help it lighting the LED on, as your problem seems to be (and NOT to turn it off).
