It's interesting to note that the homeless population in DC is at it's lowest point in several years, mostly due to a decrease in homeless families. According to the Community Partnership for the Prevention of Homelessness, there are only 690 people living on the street in a given night (with more in emergency shelters). That's roughly the amount of people who could fit into a single apartment building. For all the amount of time, money, and energy devoted to this issue, the problem doesn't appear that insurmountable.

https://community-partnership.org/homelessness-in-dc/