First, look to your lease. Second, if you did not cause the appliance to breakdown, then the landlord is probably responsible for replacement (depending on your lease). You need to send a written request to the landlord via certified mail, return receipt requested. It is possible you could sue for the cost of lost medicine in justice court (small claims court). In the meantime, you should consider buying a mini-fridge.
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If the lease is silent then the landlord is responsible for maintaining the property. But if he did not realize the refrigerator was breaking down that might make the case more difficult. See if your lease addresses repairs.
My son is a type 1diabetic so I know how expensive insulin is! Diabetes is a very expensive disease so I know that is a tough blow to lose such expensive medication. Very often in this situation a pharmacy or the manufacturer will replace insulin that goes bad. Did you ask them? If the pharmacy won't help ask your doctor's office if they can get a replacement or change your prescription so insurance will cover what you lost. Sometimes an insurance company will even do the right thing. It has happened (or so I am told).
Good luck and keep up the good fight. Diabetes is a tough disease and some days it can seem overwhelming. But just never quit and never stop your focus on control. They have made such advancements that there really is hope of achieving good blood sugar control so that diabetes' ravages are delayed and sometimes even prevented.
Don't let it get you down because being blue worsens blood sugar control and a positive attitude is half the battle.