My mother has recently been under some stress from physical illness and general anxiety. As a result, she has decided, to briefly put it, to pay some people (said to be pious, Shuyookh) in Bangladesh (home country) to recite some verses from the Qur'an. In doing so, out of this charity, she may be relieved of her anxiety. Never having heard of such a thing, I advised her not to do so, while I ask someone about it. What is the ruling on this? It may help for you to know that the thing to be recited is called 'Khatme Yunus' (or Khatme Shifa'a) the Shuyookhs will recite the verse' 'La Ilaha Illa Anta Subhanaka, Inni Kuntu Mina-Dhalimin' apparently 125000 times. (There are other verses in between). This is, as she says similar to what Yunus (as) did when he was inside the fish. She says she will not do it herself as it takes very long, and this sort of thing has been done before (she read it in books from the Indian sub-continent) payment is required as those Shuyookh, will not do it otherwise because it takes so long. Please inform me the ruling on such a thing that I may advise her.

Praise be to Allah, the Lord of the World; and may His blessings and peace be upon our Prophet Muhammad and upon all his Family and Companions.

There is no blame in treatment with al-Ruqyah based on Qur'anic passages and Prophetic supplications. So, you have to seek a pious man to do al-Ruqyah to your patient. It is Haram to go to magicians and jugglers for treatment or anything else. To be in accordance with Sharia, al-Ruqyah must meet the following three points:
1) To be through the Qur'an, Allah's attributes and Beautiful names or the Prophetic supplications.
2) To be in Arabic and in meaningful words, since it is detestable to use meaningless or unknown words while doing al-Ruqyah. However, if one does not know Arabic, one is allowed to use another language.
3) No person should believe that al-Ruqyah alone produces an effect on the patient; it is by Allah's decree.
The Prophet (Sallallahu Alaihi wa Sallam) said: "It is all right to make al-Ruqyah unless it is associated with any form of Shirk". [Reported by Imam Muslim]
As for what you named "Khatm Yunus", there is no proved evidence in Islam for that. So, you have to avoid it.
As to Dua' at the time of distress and critical situation, one is advised to say: "O Allah! There is no God but You. Glory be to You. I have been a wrong-doer".
The Prophet (Sallallahu Alaihi wa Sallam) said: "A Muslim who prays to Allah for something using the above-mentioned Dua', his/her Dua' will be answered".
Thus, you have to urge your mother to repeat this Dua' often and to seek Allah's help and Mercy to restore her health, since everything in this world is in His Hand.
Allah knows best.