Hi Nick,
If someone enters your property while you are unaware (Eg, asleep) and harms/kills themselves through substance misuse, it's not your problem.
Naturally the Police would need to investigate the death and you may well be questioned as part of that process - but that doesn't mean you would be liable for the guys death.
If you are reporting these sharps being dumped, but are not seeing any proactive police action to try and combat/reduce this, then you need to escalate this within your local Safer Community Team at West Mids Plod.
Your best bet would be to put a phone call in to Inspector Andrew Bullman, who can be contacted on 0845 113 5000 extension 7881 6229. *
He is in charge of your "neighbourhood area" (Walsall) and can allocate resources to deal with your problem. You can also ask him to confirm whether or not your issue is a neighbourhood priority, and ask what action is being taken to tackle it.
In addition, you could also have a friendly chat with him about the information given to you by the Constable. If it transpires he did give out bad info, the Insp. will be able to set him straight without dragging him through the brown stuff known as Professional Standards...
Good luck

*Disclaimer - names and contact numbers in public domain.