Is it permissible to give zakaah to a family fund to help the poor?

Whoever has money that reaches the nisaab (minimum amount at
which zakaah becomes due) and a full hijri year has passed (since acquiring
it) has to pay zakaah on it. 

This nisaab for money is
the equivalent of 85 grams of gold or 595 grams of silver, whichever is
lower in value.  

See al-Sharh al-Mumti’, 6/103-104); Majmoo’ Fataawa
Ibn ‘Uthyameen, 18/248.

Based on this, the amount that you have – 5000 pounds –
reaches the nisaab, so you have to pay zakaah. 

Secondly: The amount that you have to give is one-quarter of
one-tenth, or 2.5%. With regard to the amount in question – 5000 pounds –
the zakaah that is due is 125 pounds. 

Thirdly: If a hijri year has passed since you acquired the
nisaab, then zakaah must be paid immediately and it is not permissible to
delay it. See al-Sharh al-Mumti’, 6/186. 

Based on this, your delaying zakaah for half a year after the
full year had passed is a shortcoming and you have to repent from that and
hasten to give the zakaah. 

Fourthly: With regard to your question, “If anything is left
of this 5000 after another year has passed, will I have to pay zakaah on
it?”, the answer is: every time a year has passed and the nisaab (minimum
amount) is still there, zakaah is due on it. If the amount of money falls
below the niasaab then no zakaah is due. If the money reaches the nisaab
again then zakaah is due again, and the reckoning of the year should start
from the time when the money reaches the nisaab. 

Fifthly: With regard to giving the zakaah to the family fund,
there are eight areas in which zakaah may be spent, which Allaah has
described in the verse where He says (interpretation of the meaning): 

“As-Sadaqaat (here it means Zakaah) are only for the
Fuqaraa’ (poor), and Al‑Masaakeen (the poor) and those employed to collect
(the funds); and to attract the hearts of those who have been inclined
(towards Islam); and to free the captives; and for those in debt; and for
Allaah’s Cause (i.e. for Mujaahidoon — those fighting in a holy battle), and
for the wayfarer (a traveller who is cut off from everything); a duty
imposed by Allaah. And Allaah is All-Knower, All-Wise”

[al-Tawbah 9:60]

Giving zakaah to poor
relatives is better than giving it to others, because the Prophet
(peace and blessings of Allaah be upon him) said: “Charity given to a poor
person is charity, and charity given to a relative is two things: charity
and upholding the ties of kinship.” Narrated by al-Nasaa’i, 2582; classed as
saheeh by al-Albaani in Saheeh al-Nasaa’i.

 There is nothing wrong with giving zakaah to the family
fund, so long as you tell the person in charge of the fund that this is
zakaah, so that he will spend it in one of the areas defined in the verse
quoted above, and he will hasten to spend it immediately.  

And Allaah knows best.