You might just try using the cameras internal flash, without the umbrella. It could be a big help in this use. In the poor light, it will be much better than not using it, and it might well be all you need. It is low powered, but is all the power your use needs. Just be careful NOT to overexpose with the flash. Use TTL mode, and if it is too bright, use Flash Compensation (for example, like at -1 EV Flash Compensation) to turn it down if necessary. A few tests should make it obvious. I think you should try it first.
Perhaps you are already using it? Because you said 1/60 second, which is the automatic Minimum Shutter Speed With Flash in the auto modes (A, P, or Auto).
If the 1/60 is a concern, the way to make the shutter faster with flash indoors is to use camera M mode, then you can set it higher, but you cannot set it higher than the D40 1/500 second Maximum sync speed. Not to worry, the TTL flash still remains fully automatic flash even in camera M mode (the camera M mode is only about the indoor ambient light level). The 1/60 will not matter anyway, the flash is brighter than ambient and much faster than your shutter. The 1/60 without the flash would surely be well underexposed in your indoor case, so it has little effect.
But to answer a couple of the questions:
The Alienbees are a fine flash, but the D40 is ISO 200 minimum, and tattoos are a small area at a close distance, so Alienbees might be excessive power, overkill. A speedlight would be very adequate power for most indoor uses, and can be mounted in an umbrella (if desired). There are many great speedlights priced well less than $100, which would be very suitable.
For a remote flash in an umbrella, there are three common ways to trigger them.
One is to use a PC sync cord between flash and camera. A hot shoe adapter like the Nikon AS-15 is a good choice to add a sync cord connector to the camera without one. It is $20, and there are some half that price, but the Nikon is good and very reliable. Many speedlights already have a PC connector for this sync cord, but similar adapters (but different, NOT AS-15) can be added to their flash foot to add a PC connector.
Another way is that many speedlights (but not all) have a built-in optical slave trigger mode (or an accessory can be added) to trigger the flash in sync by the flash from the cameras internal flash (internal flash set to low power in Manual mode, which could also be fill).
Another is a radio trigger set, which is a transmitter for the camera hot shoe, and a receiver on the flash foot (or PC connector), which triggers by radio. These are more reliable in bright sun at great distance, but overkill for your indoor close use, the sync cord would be my preference.
These remote methods (mostly) are all Manual flash mode, and are NOT TTL. But your setup sounds like it would mostly always be the same lighting situation, and manual flash sounds easy.
Actually four ways, you could use the Nikon SC-28 hot shoe extension cable for the remote flash, and it could even use a TTL flash too. However it is short, it can't be more than 3 or 4 feet from the camera, because the coiled cord tension would then be strong enough to tip the light stand over. The old style SC-17 cable is plentiful and inexpensive on Ebay, and it is exactly the same as SC-28 except it does not have the hole for the new style pin lock. It works fine anyway, the pin is spring loaded, so just drop the pin to lock it anyway. That's all I have, a couple of the old SC-17 cords, and they still work fine.
FWIW, these subjects are a lot of what my own site at https://www.scantips.com is about.