There aren't really a lot of letters between European kings and the Mongols directly...mostly, there are letters between different European rulers that talk about the Mongols.

There are a few collections of medieval letters and other documents translated into English. I'm not sure about sources translated into any other languages.

A great resource is Jessalynn Bird, Edward Peters, and James M. Powell, Crusade and Christendom: Annotated Documents in Translation from Innocent III to the Fall of Acre, 1187-1291 (University of Pennsylvania Press, 2014). This book has a translated letter from the Mongol khan Hulegu to Louis IX of France

The same letter is also translated in Malcolm Barber and A.K. Bate, Letters from the East: Crusaders, Pilgrims and Settlers in the 12th-13th Centuries (Ashgate, 2010)

You said you already found letters between the Mongols and the popes, but there are some translated letters between the khan Guyuk and Pope Innocent IV in S.J. Allen and Emilie Amt, The Crusades: A Reader, 2nd ed. (University of Toronto Press, 2014)

Matthew Paris died in 1259 so his chronicle doesn't include anything about Edward I's reign, but Edward certainly did communicate with the Mongols. For more about Edward, see Jacques Paviot, "England and the Mongols (c. 1260-1330)", in Journal of the Royal Asiatic Society, series 3, vol. 10, no. 3 (Nov., 2000), pp. 305-318.

For relations between the Mongols and Europe in general, see Peter Jackson, The Mongols and the West (Routledge, 2005).