Albert, do you have a multimeter? They are pretty cheap. You can use it to determine if the bulb is good or with a bulb installed, if the unit is functioning.

There are a number of things that can be going wrong.

Spouse says I didn't do it right. Ever hear that?

To see if the bulb is working I had to reassemble the whole contraption and check for a blue light through one of the holes where water enters or exits. I had just removed the removed the bowels of the thing and replaced the bulb and attempted to test it without reassembling.

LOL, yeah, I have to leave the room or take whatever out of his hands to avoid a catastrophe or listening to a tantrum. He really does not want to be able to fix anything.

The UV is set up that way as a safety precaution. They don't want you to look directly at the light because it will damage your eyes. The port they give you to look through in most models is not a direct view.

I check mine at night, it gives a faint bluish glow. DO NOT try to make it run while open up, the UV can damage your eyes. I don't know how fast but why risk it?

I had the power brick on one UV lamp go bad so it wouldn't light even with a new bulb, but I only need it for a month or so in the spring till my plants catch up.

UV bulbs are designed to work only a limited amount of hours as prescribed by the manufacturer. Once it's Ultra-violet output drops to 60%, eventhough the bulb is still lit it is really no longer effective against algae growth or bacteria elimination. It is just "burning up" electricity. Most UV bulbs need to be replaced after 12 months of use. Some may last 18 months, but that is uncommon.

Albert,
Yes do not look at the bulb either directly or looking through the water inlet or outlet. Depending on the bulb and the distance that your eyes are from it will determine how fast it will burn your eyes. It can burn your eyes in less than a second. You may not know that you have burned your eyes when it happens. It will show up that night when you are sleeping when you wake up with your eyes burning.
Depending on the unit that you have you could be right or your wife could be right. Some units have either a pressure switch or a flow switch built in so the bulb will not come on unless the unit is operating. These are usually the better units.
Unless you check a bulb with a meter as Sandy said or can actually see a break in the filament you can not tell if the bulb is bad. Just because the bulb is black in the filament area does not mean the bulb is bad. That black color can happen in a very short period of time.
Most likely the ballast is burned out and needs to be replaced.
Bigstan is correct in that manufacturers usually rate their UV based on the output of bulb at what they determine to be the end of the life of the bulb. This is normally when the output drops to 80%. Most bulbs hit this point at 9000 hours of use. But there are other things that will affect these numbers. One thing is if the bulb is turned on and off a lot. These bulbs are designed to be turned on and left on and turning them on and off a lot will shorten the life.
You also have to look at the size of the unit, the flow rate of water through the unit and the location of the pond. As an example I have one client that has an Emperor Aquatics 120 watt high output unit on a 2000 gallon pond. This unit is so over sized for the pond that if the water flow rate through the unit is set for the pond size the output of the bulb could get down to 20 or 40% and still keep the pond from turning green. I have also seen units on ponds that were too small for the pond and would only work when bulb was close to 100% output. Then there are manufacturers that rate their units so high that they can't possibly work even when they are new.
Mike

I wonder if placing a baggy of green water and daphnia close to the light would detect every organism is killed after a few minutes under the light

Most people think that a UV light kills the organism but it does not. What it does at the rate that the organisms are exposed to the light is destroy the DNA so that the organism can't reproduce. So in your example you would see any change and the only organisms that you would see dead would be the one's that died from natural causes. You might see a little more death than normal only because of the long exposure time.
Mike