Historically players are usually not cut or released from their teams until the third substance abuse violation, there are exceptions to this though. For instance the Giants recently parted ways with Will Hill, a safety with little to no impact on the field and only played special teams. Key players either due to how important they are to the team or their cap hit stick around.
The Cardinals will not be cutting Washington for a few reasons. His cap hit would be around 7.5 million due to its acceleration and also the fact that they just paid him a bonus of 10 million when they exercised his option. The team is about to challenge the players rights to collect on that bonus saying that they should never have to pay him in the first place ( The actual bonus breaks down into two, 5 million payouts, which haven't been paid yet.)
What would it look like to the world if a team just up and cuts a player with abuse problems without trying to help them get past it? We did draft Tyrann Mathieu remember. Washington will miss 20 games total and that's before his punishment for assaulting his ex-girlfriend happens. Clearly the Cardinals are upset with him but i'm sure on the inside they are doing everything they can to help him or they would have let him go already.
The last reason the Cardinals will not cut Daryl Washington is that he is simply too good of a player to throw away, regardless of his off the field flaws. This is the same reason that the Browns will not cut Josh Gordon. In todays NFL grabbing a franchise player is near impossible and keeping them for as long as you can is a priority. Washington has problems but people forgive, we may be angry now but next year when he gets his 10 sacks and 5 interceptions this will be but a distant memory.
If you'd like to see the full list of 2014 substance abuse suspensions SBnation has a great article.