Is it permissible to keep a handkerchief stained with the blood of a girl who was killed in a car accident?

It is obligatory to safeguard the trust and keep these items that your friend entrusted to you until you return them to him as they are; as Allah Almighty says: ﴿Indeed, Allah commands you to render trusts to whom they are due﴾ [An-Nisa: 58], and as the Prophet, peace and blessings be upon him, said: «Render the trust to the one who entrusted you, and do not betray the one who betrays you.» The fact that the items include a handkerchief with the girl's blood does not affect the obligation to safeguard the trust. It is advised to reach an understanding with your wife and avoid problems.