What can I do to reduce the amount of bird dander?
Make sure that your bird is able to bathe himself. Aside from that, the rest comes down to cleaning, like changing newspaper/cage liners, sweeping, vaccuming, etc. Since your wife is the one allergic, it is best if she does not clean the cage at all, or if she does, to wear a face mask. Reducing the number of birds can help, and moving the cage to a place where there is not carpet (say, tile instead) helps a lot.
My wife just found out she has a condition which is worsened by many types of allergens, like bird dander.
As you pointed out, bird dander is only one part of the equation.
Air purification units can help, but certain units work better than others - and they only work well if their filters are cleaned.
Replace the air filter in your home (where the Air Conditioning comes in/out) on schedule, and use the highest quality pads. There is a rating that is acceptable for removing allergens.
Replace/clean the air filter in your vacuum on schedule, and/or consider a vacuum that has high-grade filters, like HEPA.
If mold is a trigger, you'll want to get rid of any indoor plants, as they are high producers of it.
Depending on allergens, you may wish to look into special bedding. There are beddings available now that have initial scientific results pointing to better health for asthmatic patients.
There are many other tips, but I wanted to cover the ones that have the highest impact
