I can't find any authentication.
That site you reference claims the Pope is an impostor.
This is not to say the letter may be authentic or not but that the site seems spurious.
I can however, find the same letter on other sites and though in all there are claims of:
St. Elizabeth, Queen of Hungary, with St. Matilda and St. Bridget, wishing to know something of the Passion of Jesus Christ, offered fervent and special prayers, upon which there appeared to them Our Lord Jesus Christ Who spoke to them in the following manner:
and
Copy of a letter of the Oration found in the Holy Sepulchre of Our Lord Jesus Christ in Jerusalem, preserved in a silver box by His Holiness and by the Emporers and Empresses of the Christian Faith.
These claims seem to contradict each other to whether the letter was found in the sepulcher and Jesus wrote it or it was dictated to the afore mentioned saints.
I cannot find any prudential coorberation of either claim which is not to say it doesn't exist but just that I cannot find it. The mispellings and type-o's in the second quote raise my curiosity as well.
On the surface, it appears to be a superstitious, and if its dated as being from the Sepulcher, possibly the first chain letter ever.
Just thinking out loud here, yet I know of know writings made by Jesus and suspect if there were they would be big news.
I find it odd that if Rome were in the possession of such a relic I would have never heard of it before or that more would be known about it and readily findable in sources with an imprimitur.
That page does cite the following at the bottom:
Benedetta DA S.S.; Pope Leo XIII, in Roma 5 Aprile 1890
This may be a hint to find an actual document but I am out of time at the moment.
Since an internet search didn't yeild anything, from here I would take the question to proper authorities.
Start a thread on CA in the traditional forum with the same question. There may be someone there who could help with finding something more substantial.
