How long should I have swollen gums after deep cleaning?

Not long. If you're following good dental hygine routine, bleeding should reduce then stop over a few wks or less. Do you floss? Often times when we don't floss regularly or brush thoroughly some tissues don't get in contact with either floss or toothbrush, & become inflamed/irritated and gum disease can develop. A water pik might help too. Be sure to see your dentist regularly, esp if it doesn't clear.