What are some quick procedures that can successfully remove pet dander?

Real estate agent. Find one. Buy new house. Leave pet behind. There are no other "quick procedures". Moving pet out is your best solution in long run. Keeping pet off furniture, bedding & out of laundry room helps. A hepa air purifier reduces airborne pet dander as will keeping windows open. For effective hepa calculate volume of home. Hepas must move that amount of air every 10 minutes. It's a big investment.
Bathing. If you can do nothing else, bathing the cat or dog, ideally twice per week, will help the most. Hard floors, vacuuming with a filter and using a hepa filter in the room will also help. Nothing eliminates pet dander though, other than relocating the animal plus time and cleaning.