"Never answering the phone! Now missed appointment."

This isn't so much of a complaint but I have no idea how to get hold of this department! My last appointment was a couple of months ago and I was give a sheet of paper that told me when my next 2 appointments were. I realized I'd lost this piece of paper last week but I knew the appointments were in June. So last week I started calling the number it says on my last letter and nobody answered. I called multiple times a day on multiple days. Then on Monday I was called by a lady in dental appointments and she said she was calling to remind me that I had a dental appointment on the 16th (one which I knew about). So I asked her if she could tell me the dates for my orthodontic appointments and she informed me that I had one on the 4th which I'd missed. Since she informed me of this I have tried to call so many times each time I finally reach 'dental appointments' and they connect me to orthodontics and it just rings indefinitely. I still have not got hold of this department having once been told that all their lines are busy (how can they be if nobody ever answers the phone?) and once that they were gone for lunch (which is fair enough) and so many times it just rang and rang until I decided to hang up and not waste anymore time. The outcome is that I've now received a letter telling me that I missed my appointment and that I won't have any further appointments and if I want to discuss this I have to contact them in writing! I feel a bit aggrieved because I absolutely tired my hardest to get hold of them and ask when my appointment was and I couldn't even get through to 'dental appointments' at first. Now I've missed it and might not get my treatment because the medical staff probably think I was too lazy to attend the appointment or call up to speak to them. My other issue is that one lady kindly phoned me to remind me of my appointment that is on the 16th so why didn't anybody do the same for my one on the 4th? Some consistency would be nice.