Thank you very much to caring reader, Hava, for this update and for letting me know that the father still needs davening & for providing his name.
Also, please daven for her newly born baby, Rach Hanolad ben Shira Yael.
While the medical staff was able to deliver the baby prematurely, both he and his mommy are still in critical condition.
The baby has been given an especially harsh prognosis, but you don't need to believe it because prayer can change everything.
May they have a complete and speedy refuah among all the other ill Jews.
According to the Chida, the following Tehillim/Psalms can be helpful:
9 is good to say for a sick child.
22 is good for every kind of trouble (tzaar)
92 can help one merit seeing great miracles.
116 can save one from a strange death.
119:89-96 (lamed) sweetens or nullifies a negative judgment.
127 is good to say for a newborn.
Also, 119 is divided up according to Alef-Bet and there's an old tradition that it's very good for a sick person if you recite the sections according to their name.
In this case, you have:
שירה יעל בת ליאורה שרה
So you would recite the following:
119:161-168 (shin) ש
119:73-80 (yud) י
119:153-160 (reish) ר
119:33-40 (hei) ה
119:73-80 (yud) י
119:121-128 (ayin) ע
119:89-96 (lamed) ל
119:9-16 (bet) ב
119:169-176 (taf) ת
119:89-96 (lamed) ל
119:73-80 (yud) י
119:1-8 (alef) א
119:41-48 (vav) ו
119:153-160 (reish) ר
119:33-40 (hei) ה
119:161-168 (shin) ש
119:153-160 (reish) ר
119:33-40 (hei) ה
And do likewise for Rach Hanolad ben Shira Yael:
רך הנולד בן שירה יעל
119:153-160 (reish) ר
119:81-88 (kaf) כ
119:33-40 (hei) ה
119:105-112 (nun) נ
119:41-48 (vav) ו
119:89-96 (lamed) ל
119:25-32 (dalet) ד
119:9-16 (bet) ב
119:105-112 (nun) נ
119:161-168 (shin) ש
119:73-80 (yud) י
119:153-160 (reish) ר
119:33-40 (hei) ה
119:73-80 (yud) י
119:121-128 (ayin) ע
119:89-96 (lamed) ל
You can also go to the following site and type in the names in Hebrew and it will give you the sections you need from Tehillim 119 in the right order (instead of flipping around in your Tehillim book). You can print it out too:
tehillim-online.com/read-segula/psalm-119-by-the-name
There are other traditions for saying Tehillim by the letters of the ill person's name, but the above tradition is the one with which I'm most familiar.
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May Hashem please bring Mashiach now in a sweet way.