Is it permissible for her to inject urine for medicinal purposes?

Firstly: 

Human urine is najis (impure) according to scholarly consensus. The principle that is agreed upon – in general terms – is that it is not allowed to use impure substances for medicinal purposes, because of the report narrated by Abu’d-Darda’ (may Allah be pleased with him) who said: The Messenger of Allah (blessings and peace of Allah be upon him) said: “Allaah has sent down the disease and the remedy, and He has created for every disease a remedy, so treat disease but do not treat it with anything that is haraam.”

Narrated by Abu Dawood, 3874. See also al-Mawsoo‘ah al-Fiqhiyyah, 11/119 

Secondly: 

The scholars differed concerning cases where one is compelled by necessity to treat disease with haraam things: is it permissible to drink urine and other impure substances for the purpose of treating disease? Or is it permissible to treat disease with impure substances when it does not involve drinking it? Or is it is not permissible in all cases? In al-Mawsoo‘ah al-Fiqhiyyah (11/119) it says:

The Maalikis regarded this ruling [the prohibition on treating disease with najis and haraam substances] as being general in application, including everything that is impure (najis) and haraam, whether it is alcohol, dead meat (from an animal that was not slaughtered properly) or anything else that Allah, may He be exalted, has forbidden, and whether the means of treating disease is by drinking it or applying it to the body, and whether it is used on its own or mixed with a permissible medicine. They made one exception to that and regarded it as permissible to use it for treating disease, which is when the method of treatment is by applying it to the body and there is the fear that not doing it would lead to death, whether the substance applied is najis or haraam, used on its own or mixed with a permissible medicine. 

The Hanbalis added to the category of haraam and impure (najis) everything that is regarded as off-putting, such as the urine of animals whose meat may be eaten and others, except the urine of camels, which it is permissible to use for medical treatment. End quote. 

See: al-Majmoo‘ Sharh al-Muhadhdhab by an-Nawawi, 9/54; Radd al-Muhtaar by Ibn ‘Aabideen, 6/389 

Thirdly: 

Using haraam and impure substances in ways other than eating or drinking them, whether that is by means of injection, application to the body, and so on, is less serious than drinking or eating the impure substance. Hence there are some scholars who granted a concession allowing that, even though they did not allow drinking impure substances for the purpose of medicinal treatment. 

It says in al-Mawsoo‘ah al-Fiqhiyyah (2/88): With regard to injections using haraam substances, the scholars said it is not allowed unless it is necessary, because of the general meaning of the prohibition on using haraam substances. 

But if the injection is necessary, then the Hanafis and Shaafa‘is allow injection because of necessity, if the necessity is because there is fear for the patient’s life and a skilled Muslim doctor has told him that his recovery is dependent on treatment with the haraam substance, on condition that he use only as much as is necessary. This was confirmed by Ibn Hazm. 

The Maalikis say, and this is also an opinion of the Hanbalis, that it is not permissible to apply to the body, inject or use for medicinal purposes alcohol and impure substances, even if refraining from doing so may lead to the patient dying, because the Prophet (blessings and peace of Allah be upon him) said: “Allah has not put healing for my ummah in that which He has forbidden to them.” And when mention was made to the Prophet (blessings and peace of Allah be upon him) of nabeedh made for medicinal purposes he said: “It is not a remedy; rather it is a disease.” End quote. 

What appears to be the case, and Allah knows best, is that the issue of inserting urine or injecting the patient with it through the vagina in the case of necessity is less serious, and the concession in such cases is broader, but that is subject to conditions: 

1.That it be on the advice of a Muslim doctor of good character who is trustworthy and skilled, and he knows that this will lead to recovery.

2.That there is no permissible remedy that could take its place.

3.That it be used only as much as is necessary.

An-Nawawi (may Allah have mercy on him) said: Our companions said: That is only permissible if the one who wants to apply this treatment to himself is well versed in medicine, and he knows that there is nothing else that could take its place, or he is told about that by a Muslim doctor of good character; and one doctor’s opinion is sufficient. … If the doctor told you that it will speed up recovery, and that if it is not done recovery will be delayed, then there are two opinions as to whether it is permissible. Both opinions were narrated by al-Baghawi and he did not regard one of them as stronger than the other. But the more correct opinion is that it is permissible.

End quote from Sharh al-Muhadhdhab, 9/54 

To sum up: there is nothing wrong with your friend using urine to treat her condition according to what the doctor has told her, if the doctor is trustworthy and there is no other permissible alternative that could be used instead. 

And Allah knows best.