It's so very, very, very hard. Make sure you look after yourself, remember to eat and sleep yourself, etc! Do you have any real life suppot?
My dad decreased the amount he ate over the last few years of his 10 year journey with dementia, and in the last month or so barely wanted to eat anything - he would have occasional tiny cheese sandwiches, milk in his tea, and still sometimes liked puddings, like custards or ice cream. That was it.
Have you completed a RESPECT form? If not, I urge you and his other close loved ones to speak to the care home about this ASAP, so that you can plan an end-of-life journey for your dad that reflects what his wishes would have been when well (if you see what I mean).
Pretty much straight after Xmas this year, my dad (who was suffering from repeated chest infections at this time also - he would be hospitalised, given anti-biotics, but as soon as he finished the course the infection would return) stopped eating and drinking, and we made the decision with his GP to withdraw all but palliative care, knowing this would lead to his death within weeks at most. It was his time.
My mom, brother and I cared for him at home, with carers coming in twice daily and the district nursing team available to administer morphine and Valium (bascially we had to call them if he was distressed, and they would arrive within the next two hours - it's not an ideal set up, but there you go).
This stage of his end of life lasted 10 days. I had not realised someone could last so long without drinking, but apparently as the body shuts down it also requires less.
My brother in particular struggled with not giving him liquid, but the nurses told us that at that point liquid but would have gone onto his lungs and cause him pain.
It was the right decision to allow him to pass at this time, and was what he would have wanted. In some ways I really treasure those days we had together as a family and getting to say goodbye to dad - I spent a huge amount of time next to him on the bed as my presence seemed to comfort him - on the other hand, watching someone you love so much slowly die like that is traumatising (not a word I use lightly) and I think we all still have the mental scars.
Everyone's end of life is different, as it should be - but you have any questions at all please feel free to ask, and I will answer as fully as I can.