Praise be to Allah.
The part that is saheeh of the hadeeth mentioned by the questioner is “Treat your sick ones with charity.” This was classed as hasan by Shaykh al-Albaani in Saheeh al-Jaami’ (3358). But some scholars classed it as da’eef (weak) and did not think it was saheeh or sound.
As for the extra material that was mentioned by the questioner, “and purify your wealth with zakaah,” al-Albaani said concerning it in al-Silsilah al-Da’eefah (3591): It is mawdoo’ (fabricated).
The same applies to the additional material that follows it: “and respond to calamity with supplication and beseeching of Allaah.” Al-Albaani said in Da’eef al-Targheeb wa’l-Tarheeb (456): It is da’eef (weak).
There is nothing wrong with the sick person sacrificing a sheep or the like in the hope that Allaah may heal him thereby from the pain and sickness he is facing, because charity is one of the means of healing.
See the answer to question number 107549, where we quoted fatwas from the Standing Committee for Issuing Fatwas and Shaykh ‘Abd al-Rahmaan ibn Jibreen concerning that.
But it should be noted that the charity mentioned in the hadeeth does not refer specifically to offering a sacrifice; rather it includes giving food, clothing, money and so on in charity.
The supplication that the questioner mentions is not narrated in the Sunnah -- as far as we know -- so he should not adhere to that and believe that it has any superiority over any other supplication.
It is prescribed for the believer to pray that his righteous deeds be accepted. “And (remember) when Ibraaheem (Abraham) and (his son) Ismaa’eel (Ishmael) were raising the foundations of the House (the Ka‘bah at Makkah), (saying), ‘Our Lord! Accept (this service) from us. Verily, You are the All-Hearer, the All-Knower’”
And Allaah knows best.