Sorry for really late response.
Does OP's "Mongolian Russia" means only the Golden Horde? Or does it also include the principalities under the (indirect) rule of the Golden Horde?
Notes also that I read neither (Old) Russian nor Mongolian-Turkic, nor Arabic by myself so that my knowledge on that field of research is really limited.
As for the Golden Horde itself, I unfortunately know very little.
On the other hand, as for the Russian principalities:
Zenkovsky, Serge A. (ed. & trans.). Medieval Russia's epics, chronicles, and tales. rev. ed. New York: Dutton, 1974 (1st ed. 1963): includes some basic narrative texts from Kyivan/ Kievan Rus' to Early Modern Period. As for so-called Tatar Yoke period, a few, mainly religious texts are included, but also Zadonschchina, one of the primary texts narrating the battle of Kulikovo (1380).
Complete Collection of Russian Chronicles/ PSRL: Polnoe Sobranie Russkikh Letopisey is the series of medieval Russian chronicles, but only a few of these chronicles in addition to PVL (Primary Chronicle) have been translated into English, or the translations are sometimes difficult to access.
The Chronicle of Novgorod 1016–1471, trans. Robert Mitchell & Neville Forbes, London: Royal Historical Society, 1914: is a convenient English translation uploaded online, but I'm not so sure about the standard of textual criticism of this old translation.
The Galician-Volynian Chronicle: An Annotated Translation, trans. George A. Perfecky. Harvard Series in Ukrainian Studies, vol. 16, II : The Hypatian Codex, part 2. Munich: Wilhelm Fink, 1973: is academically critical edition, but I'm afraid it might be difficult to find unless OP is affiliated with the decent university (and its library).
Simeonovskaya letopis' [The Simeonovskaya chronicle] (The original is to be found in PSRL 18) also contains some information on the relationship between the Golden Horde and Russian princes, but I don't think the English translation is available for this text.
English translation of the 16th century Nikonian Chronicle by Serge A. Zenkovsky & Betty J. Zenkovsky in multiple volumes (vols. 3-4 cover from the 13th to 15th centuries) is also published in the last decades of the 20th century, but I'm rather careful of the source value of this text for earlier period.
(added): Prof. Emeritus Daniel Kaiser, one of few Anglophone authority on medieval Russia, also uploads some English translation of medieval Russian law and treaties in his site: http://web.grinnell.edu/individuals/kaiser/Trans.html
Apparently Recommended Reading (to check the bibliography section/ I don't have a copy to check by myself, sorry)