No matter to whom Peter's letter was originally intended, GOD preserved it & made it become Scripture, available to all. The facts in that letter are GOD'S, & He made sure they were available to the whole world.
James 5 has some facts and some promises concerning sickness. He told them what to do to get well and it worked every time it was tried because the conditions for success were built in. However, during that same time period Paul, who had probably read the letter from James (James was the first NT epistle written in 45 AD) would know about his cure at the time of this important incident.
Phil 2:24 But I trust in the Lord that I also myself shall come shortly.
25 Yet I supposed it necessary to send to you Epaphroditus, my brother, and companion in labour, and fellowsoldier, but your messenger, and he that ministered to my wants.
26 For he longed after you all, and was full of heaviness, because that ye had heard that he had been sick.
27 For indeed he was sick nigh unto death: but God had mercy on him; and not on him only, but on me also, lest I should have sorrow upon sorrow.
28 I sent him therefore the more carefully, that, when ye see him again, ye may rejoice, and that I may be the less sorrowful.
29 Receive him therefore in the Lord with all gladness; and hold such in reputation:
30 Because for the work of Christ he was nigh unto death, not regarding his life, to supply your lack of service toward me.
Philippians was written in 60 AD.
You got to ask yourself; why didn't Paul practice the James expedient? And what about this;
2 Timothy 4:20
Erastus abode at Corinth: but Trophimus have I left at Miletum sick.
Why didn't Paul heal him? He said the following under inspiration.
2 Cor 1211 I am become a fool in glorying; ye have compelled me: for I ought to have been commended of you: for in nothing am I behind the very chiefest apostles, though I be nothing.
12 Truly the signs of an apostle were wrought among you in all patience, in signs, and wonders, and mighty deeds.
What are the signs of an apostle?
Mark 16:17 And these signs shall follow them that believe; In my name shall they cast out devils; they shall speak with new tongues;
18 They shall take up serpents; and if they drink any deadly thing, it shall not hurt them;
they shall lay hands on the sick, and they shall recover.
19 So then after the Lord had spoken unto them, he was received up into heaven, and sat on the right hand of God.
20 And they went forth, and preached every where, the Lord working with them, and confirming the word with signs following. Amen.
HEAL THE SICK??? He left Trophimus at Miletum sick. Epaphroditus was sick nigh unto death and they did not practice the James 5 treatment and Paul did not heal him, God did.
There has to be an answer for these questions, and there is. It takes rightly dividing the scriptures and letting words mean what they say, and drawing conclusions from a literal understanding of words that are in a historical as well as a theological context.