Kakuzu killed him cuz they showed him holding first's neck in the battle
we know Hashirama divided his bijuu collection between the ninja villages at the moment to sign peace after Shinobi World War I, which makes Mizukage II and Tsuchikage II unlikely to be Hashirama's killer... so a missing nin could likely kill hashirama and the perfect candidate would be kin and gin bro and kakuzu as you stated it.... but
In databook 1, they say Tobirama took over the command of Konoha after his elder brother's death, so Hashirama must have been dead by the time the KinGin brothers attacked Raikage II and Hokage II. and that leave us to your BET KAKUZU..
but then again since he only claimed he fought him and didn't claimed he killed him, the odds are low that he took his life. Although it is possible that Hashirama make it out of the battle barely alive, to die a few days latter, Kakuzu not knowing how badly he injured his adversary.