The typical gain of a 2N3904 is 200-300 at a couple mA collector current (more as it warms up due to not being saturated)
Even with 560K, that's half a mA or so, which will give plenty of light from a modern LED, but you should be able to see that it's not as bright as when a 10K resistor is used.
Do not use the hfe for this calculation if you want the transistor saturated hard on, use a forced beta of something like 20 to 50, if the typical hfe is 200 or so and the minimum 100 or so. If you use, say 30, in your equation you get a resistor value of 8.8K, so you might use 10K or 8.2K.
The reason is that you won't likely have a guaranteed hfe for the current you're using, and the hfe decreases at temperature extremes. It's still only "wasting" a few percent of the LED current, so no big deal.
To prove this to yourself, take a voltmeter and measure Vce of the transistor when it is on. If it is something like 50-100mV it is saturated.
The formulation is correct. You must be careful with the \$h_{fe}\$ value you use. Search the device data sheet curve \$h_{fe}\$ vs. collector current, to the value of \$h_{fe}\$ in saturation state.
EDIT: from the datasheet
You must use the lower curve to size the output current and by the ratio
$$
\dfrac{I_C}{I_B}=1000
$$
obtain the required value of base current.
For the saturation condition, the value of \$h_{fe}\$ is lower than for the condition of linear operation, so may the value selected for the base resistance is rather high.
Best Answer
If you want to conduct up to 150mA of current between node 1 and ground and only drop 1.0V (it's not a perfect switch) then you'll need to assume the current gain is typically 50. If you want a lower saturation voltage then the spec sheets says feed the base with 15mA i.e. the current gain has dropped to only 10 but saturation will only be 0.3V.
So assuming you are happy with 150mA while saturating the transistor to about 1V, you need to push \$\dfrac{150mA}{50} = 3mA\$ into the base.
The base voltage will need about 0.7V so the remainder (1.8V - 0.7V) needs to be across the resistor R1. Ohms law tells us that R = \$\dfrac{1.1V}{3mA} = 366.7\Omega\$.
So choose maybe a 360 ohm resistor.
If this isn't good enough for your needs look for an N channel MOSFET with low \$V_{GS(threshold)}\$ - something like 1V or less.