Is it permissible for her to listen to music whilst exercising?

There is
nothing wrong with a woman doing exercise if it is in a place that is only
for women, where men cannot see them, so long as that is done to a limited
extent and it does not distract her from something that is obligatory, and
it does not lead to her missing prayers or delaying them until their time is
over, or neglecting other duties. 

Among the
haraam things that may accompany exercise in some gyms is using music and
listening to it. Al-Bukhaari narrated that Abu Maalik al-Ash’ari (may Allaah
be pleased with him) said: The Prophet (peace and blessings of Allaah
be upon him) said: “There will be people among my ummah who regard as
permissible zina, silk, alcohol and musical instruments.” 

From this
hadeeth the ruling on musical instruments is clear, which is that it is
haraam. The fact that they are mentioned alongside other haraam things,
namely silk, alcohol and zina, reinforces the fact that they are haraam. 

There is no
difference of opinion among the four Imams that musical instruments are
haraam. 

Shaykh
al-Islam Ibn Taymiyah (may Allaah have mercy on him) said: 

Whoever
plays these instruments as a form of worship is undoubtedly misguided and
ignorant. But if he does it as a form of entertainment, then the view of the
four imams is that all musical instruments are haraam. It is proven in
Saheeh al-Bukhaari and elsewhere that the Prophet (peace and
blessings of Allaah be upon him) said that there would be among his ummah
those who regarded zina, silk, alcohol and musical instruments as halaal,
and he said that they would be transformed into monkeys and pigs. 

Majmoo’
al-Fataawa (11/576, 577). 

Music
destroys the heart and distracts people from the truth; it generates
hypocrisy in the heart. It cannot soothe the nerves or be a remedy. 

Shaykh ‘Abd
al-‘Azeez ibn Baaz (may Allaah have mercy on him) said:

 Music and
other kinds of entertainment are all evil, but they are things that the
Shaytaan makes appear attractive and enjoyable, and he calls people to them
so as to distract them from the truth by means of falsehood, and so that he
can lead them away from that which Allaah loves, towards that which Allaah
hates and has decreed haraam. Music, the ‘ood (a kind of stringed
instrument) and all musical instruments are an evil and it is not
permissible to listen to them. It is narrated in a saheeh report that the
Messenger of Allaah (peace and blessings of Allaah be upon him) said:
“There will be people among my ummah who regard as permissible zina, silk,
alcohol and musical instruments.” End quote. 

Majmoo’
Fataawa al-Shaykh Ibn Baaz (3/346).

 The Shaykh
also said: 

As for using
music as a remedy, there is no basis for this, rather it is the work of
fools. Music is not a remedy, but it is a disease. It is a kind of
entertainment which cause sickness in the heart and leads to deviation of
morals. The beneficial remedy that can calm the nerves is making the sick
people listen to recitation of Qur’aan, beneficial exhortations and
beneficial hadeeth. Using music as a remedy is something that gets people
accustomed to falsehood and makes them even sicker. It makes listening to
the Qur’aan and Sunnah and useful exhortations burdensome for them. Laa
hawla wa laa quwwata illa Billaah (there is no power and no strength
except with Allaah).” End quote. 

Majmoo’
Fataawa al-Shaykh Ibn Baaz (9/429) 

See also the
answers to questions no. 43736,
5000 and
5011 for more information.   

Based on
this, you should advise these people and tell them that listening to music
is forbidden in Islam and that there is no need for this haraam deed. There
are many gyms which pay attention to that and they avoid playing music,
which makes more people come to them. If they do not respond, then at least
ask them not to use it whilst you are in the gym, so that you will not be a
partner in this sin, remaining silent about an evil which you are obliged to
denounce. 

If you
cannot do that, then you have to leave this gym: either look for another
one, or look for another solution such as buying some equipment to use for
exercising at home, which is better for you.