She neglected the birds until they died – has she committed a sin?

Firstly: 

There is no basis for the belief that it is not permissible
for a woman to go near the birds during her period. There is no connection
between the two matters. What you must do is give up such beliefs. There are
certain rulings which apply to the menstruating woman but such matters are
not among them and are not at all connected to them. 

The scholars are unanimously agreed that it is permissible
for a woman to slaughter animals (for food) when she is menstruating. See
al-Mughni, 13/311. 

Secondly: 

Everyone who keeps these
birds or any other creatures is obliged to take good care of them. If he is
not able to do that then he should not keep them, and he should sell them or
slaughter them if they are the kinds of animals or birds that can be eaten.
There is a stern warning issued to the one who keeps an animal and does not
feed it or give it water until it dies. 

Ibn Qudaamah al-Maqdisi (may Allaah have mercy on him) said: 

Whoever owns an animal must take care of it and spend on
whatever it needs of food or appoint someone else to look after it, because
of the report narrated by Ibn ‘Umar, who said that the Prophet (peace
and blessings of Allaah be upon him) said: “A woman was punished because of
a cat which she detained until it died of starvation, because she did not
feed it or let it go and eat of the vermin of the earth.” Agreed upon. If he
refuses to spend on it, he should be forced to do so; if he refuses or is
unable to do that, he should be forced to sell it or to slaughter it if it
is an animal that may be slaughtered (for food). End quote.

Al-Mughni, 8/205 

Al-Shawkaani said, commenting on the hadeeth about the woman
who detained the cat: 

It is understood from
this hadeeth that it is haraam to detain cats and similar animals without
giving them food and water, because this is tormenting the creation of
Allaah, which has been forbidden by the Lawgiver… al-Nawawi said: It seems
that she (this woman) was a Muslim, but she entered the Fire because of this
sin. End quote. 

Nayl al-Awtaar, 7/7 

It says in al-Mawsoo’ah al-Fiqhiyyah (5/119, 120): 

It is permissible to keep
animals for beneficial purposes, such as guarding (property etc), listening
to their voices and enjoying looking at them, but the one who keeps them is
obliged to give them food and drink, because of the sanctity of life. End
quote. 

In the answer to question no. 48008 it says: 

It is permissible to keep birds and similar creatures in
cages for pleasure, especially if that is because of their appearance or
their voices, subject to the condition that one feeds them and gives them
water. End quote.   

See question no. 48008 for more detailed information. 

Thirdly: 

If your excuse for not feeding these birds until they died is
that you forgot and did not do it deliberately, then this is an excuse which
is acceptable according to sharee’ah. We ask Allaah to forgive you for that.
Allaah says (interpretation of the meaning): 

“Allaah burdens not a person beyond his scope. He gets
reward for that (good) which he has earned, and he is punished for that
(evil) which he has earned. ‘Our Lord! Punish us not if we forget or fall
into error…’”

[al-Baqarah 2:286]

And Allaah has answered
this prayer of the believers and says, “I have done that,” as was narrated
by Muslim (126) from the hadeeth of Ibn ‘Abbaas (may Allaah be pleased with
him). 

And Allaah says (interpretation of the meaning): 

“And there is no sin on you concerning that in which you
made a mistake, except in regard to what your hearts deliberately intend”

[al-Ahzaab 33:5]

Since you did not deliberately intend to kill them, there is
no sin on you, in sha Allaah. 

And Allaah knows best.