What should she do with her husband who is committing adultery and his mistress is pregnant?

Firstly:

It is well known in Islam that zina (unlawful sexual relationship) is haraam
and is a major sin. The one who has fallen into this sin has to repent to
Allah, may He be exalted, before it is too late. The prohibition of this evil
deed is more emphatic in the case of a married man, hence the punishment for it
is stoning to death.

See the answer to question no. 97884

If it is proven to you that your husband
is committing adultery, either because he has admitted it to you or there is shar‘i evidence to that effect, then advise him to fear
Allah, immediately end his sinful relationship with that woman, do a lot of
righteous deeds, and seek good company who will encourage him to do good and
obey Allah and to avoid evil and sin.

Secondly:

It is not permissible for a woman to
marry a man who is known to have committed zina
unless he repents sincerely. If a woman marries a man who has been committing zina, then she is sinning and her marriage contract is
invalid. If a woman marries a man who is chaste, then after marriage he falls
into zina, the marriage contract is not annulled by
his falling into zina, but this does not mean that
the wife should accept him as a husband if he does not give up this immoral
action. Hence we think that in your case, if he does not give it up, you should
not remain in the marital relationship with him. Rather you should hasten to
end your marriage to him, by means of either talaaq
or khula‘.

You should understand that if your
husband continues to commit zina, it will have a
negative impact on his family members, his wife and children, in terms of both
their upbringing and their health. So do not think of staying with him if he
does not stop and give up this sin.

See the answer to questions no. 101771 and 110141.

Thirdly:

As you are living in a non-Muslim
country and the law in this country forces the adulterer to acknowledge the
child and also to spend on the child and take care of him, in this case Islam
does not oblige you to look after the child, take care of him or breastfeed
him, even if he was your husband’s child from a permissible marriage, unless
you do that voluntarily and out of kindness.

Ibn Qudaamah (may Allah have mercy on him) said:

The husband does not have the right to
force her (his wife) to breastfeed his child from another woman, or to look
after him.

Al-Mughni, 9/313

It was narrated from Jaabir
ibn ‘Abdillah (may Allah be
pleased with him) that the Messenger of Allah (blessings and peace of Allah be
upon him) said: “Have you got married?” I (Jaabir)
said: Yes. He said: “To a virgin or a previously-married woman?” I said: To a
previously-married woman. He said: “Why not a young girl,
and you could play with her and she with you?” I said: I have sisters, and I
wanted to marry a woman who would be able to bring them together, comb their
hair and take care of them.

An-Nawawi (may
Allah have mercy on him) said:

This indicates that it is permissible
for a woman to look after her husband and his children and dependents if she
agrees to do so, but if she does not agree to do so, then no (she should not be
forced to do it).

Sharh Muslim, 5/203

Wali ad-Deen al-‘Iraaqi (may Allah have
mercy on him) said:

This indicates that it is permissible
for a woman to look after her husband and his children, sisters and dependents,
and that there is nothing wrong with a man asking his wife to do that, but it
is not obligatory for her to do so; rather she may do that if she agrees to.

Tarh at-Tathreeb, 7/112

To sum up our advice to you:

If your husband has not repented and
given up zina, then you should separate from him and
leave him and his child. But if he has repented from that and you think that he
regrets what he has done, and you think it most likely that he has mended his
ways, then there is nothing wrong with you staying with him and we advise you
to help him to look after this child and take care of him, in the hope that
Allah may reward you for that and compensate you with good, and perhaps this
child may become righteous instead of being left under the care of disbelievers
and being raised by them.

And Allah knows best.