What a terrible situation you are in! Not just dealing emotionally with you dad being seriously ill, but also carrying all these burdens that normally should belong to an adult and not a 15 year old. I am sure you are doing a very good job, which may be why your dad doesn't see the need for you to have help.
This MUST change, as you are clearly very tired and emotionally drained. It must be very hard for you to go against your dad's wishes about getting help, but I'm sure that he would agree with you if he fully understood what a burden this is to you. He has got lots on his own mind as well, and he might not have noticed quite how worn down you are. Maybe you could get someone else to talk to him about this, as this should not rest on your shoulders.
I don't know if there is any other support around you, an adult that you trust in the family or maybe a family friend or a parent of your best friend? School teachers are also good in these situations, but may not be an option now during school holidays. If your dad has regular contact with a hospital or hospice, there might be someone there you could talk to about these things (doctor, nurse, Macmillan nurse, social worker etc.)
There are probably also places you could go in your local area and I advise you to have a look at the riprap website in the section called 'support in my area'. Most cities have specific projects for young carers and you can check out if there's any near you here.
If you need someone else to talk to you could contact ChildLine, a free, 24-hour helpline for children and young people in the UK. You can call the helpline on 0800 1111 about any problem, at any time - day or night.
Your specific question is about what you can do in the last few months of his life that would make it easier for you and him. Getting help is the most crucial factor for both of you. You need adults to talk to and unload some of your responsibilities and your dad needs health professionals that can deal with his illness and the symptoms that he probably has from that. This will free up some of your time and energy so that you can talk to each other about other things than problems and maybe you can do things together as a family, depending on how your dad is feeling.
I am concerned about how you are feeling right now - stressed, depressed and not eating properly. This may all improve once you start sharing your problems with other people and hopefully get some help. I am glad that you are taking this seriously though and have taken a first step in talking to us. Unfortunately, it seems as if no-one is approaching you to help you and it is a pity that you are the one who has the burden of setting the wheels in motion. When you feel down it is easy to just let things slide and it can be very hard to make an extra effort in changing things. You have done very well in writing to us and I hope you will find the strength to make contact with people nearer you who will be able to help you further. Good luck with it all and please get back to us if there is anything else we can do for you.