I might start with "Thank you for your kind thoughts of my dad during the holidays." From there, you could say "I am sorry to inform you of his death this past summer." If you feel like you owe an explanation of not informing them sooner, you might say, "My apologies for not informing you and many others much sooner, but the sorry I have felt has kept me from reaching out." And again thank them.
Hope this helps.
Hope this helps.