It's really really common for people to feel depressed and worse after a diabetic diagnosis. (Every time I see a nurse for diabetic care I get asked if I'm feeling OK/depressed etc etc and I've been diabetic for 49 years! I'm not depressed, because I've learnt to live with the condition, and it doesn't really stop me doing anything, but I've had periods in the past when I've not been happy.)
Luckily I've not experienced the self harming thing but I know that it's not uncommon and many (particularly young) diabetics go through the diabulimia thing (eating disorder where you control your weight by skipping insulin.) As you've already experienced a DKA you'll realise what a bad idea that is.
So, you're really young. I think it's much harder for young people to cope with depression because you haven't got as much experience of it (Not saying that young people don't have bad things happen, just that they usually haven't had decades of it.). I was badly bullied when I was 14 and I thought it would never get better, I remember despairing because I thought this. After several months things improved, and by the time I was 16 the bullying had ended, so after that I always had those feelings as a benchmark of how awful you could feel and still come out of it at the other end.
But, back to the diabetes. Firstly, well done for reaching out for help. You are not alone. And it's really good that you've got some friends at home that you can talk to about it.
My workplace know about my diabetes but I don't test my sugars or inject in front of anyone (not even my family) because I feel like I'm being a nuisance when I'm injecting
Injecting and/or testing is not being a nuisance. Would you mind if one of your friends or colleagues used an asthma inhaler in front of you? Unfortunately high and low blood sugars do bad things to your mental health as well as your physical, so my recommendation is that you try to learn to cope with injecting and testing in front of other people. It's bad enough having to do it, trying to find a private place and/or time to test/inject is just piling on the difficulty.
Is there anyone you could talk to at work about your worries? A colleague, if not your manager? I'd be very surprised if anyone was anything other than sympathetic to you, and it's a good idea for your workplace to know what to do if you get hypo. (Speaking as someone who once had her life saved by a work friend who saved me from a hypo.)
Anyway, lots of virtual hugs. Remember that managing diabetes is a marathon not a sprint. It's particularly hard for the first few months/years because there's just so much to get the hang of. But you'll always find someone to listen and talk on these forums. (Though probably not for a few hours now, as most of the UK folk are in bed. I'm in New Zealand, so it's day time for me.)