This approach would definitely be more effective than our current approach to dealing with drug addicts: put them in prison. Supervising addicts and making sure that they don't do too much damage to themselves seems like a very good start. In addition to that, there needs to be some form of rehabilitation to hopefully help them get rid of their addiction.
I don't know much about the so-called "fix rooms" so I'm not sure how effective they are in looking after addicts, but Denmark must have evidence to show that it works otherwise they wouldn't keep them going. If they work for one country then why not try them out in the U.S. and see how it goes.
The number of deaths from overdoses are ever increasing in the United States. Having this type of facility that an addict could go to and get their fix may help to decrease the number of overdose related deaths. If addicts were able to be supervised while they are injecting drugs they would be less likely to over use the drug and they would have help nearby if something went wrong. That being said, use of these facilities should be limited and monitored, so that the benefit is not abused.
We know that people are going to do drugs in the United States, and we know that just fixating on the illegality is not working to curtail harm from drugs. At least in the fix rooms people are supervised and there is someone there to help make sure it is as safe as possible. The US should try this, it would help.