1) Have the Kazar khangate more thoroughly adopt Judaism, instead of keeping the oldways partially alive, and then They Conquer the Rus instead of the other way 'round. Overtime the Rus will assimilate and become Culturally and Relgiously Jewish if not Ethnically so. meanwhile the Ethnic jews intermarry with the Descendants of Converts, increasing the number of at Least Partially Ethnic Jews (often enough with Norse/Rus, Slavic and Turkic Ancestry as well. By the mid XIXth Century you're talking about a Kazar-Jewish state the size of the Russian Empire with a Similar Population. Which in 1866 had an estimated population exceeding 180 Million. Even if only Half are Jews, that's still 90 million Quasi-Ethnic Jews, with the remainder worldwide easily racking up another 10+Million, and Ashkenazim/Kazarim Jews would easily be one of the largest nation on Earth, save for the Han Chinese and the Japanese.
2) Have Ghengis Khan and the Mongol Horde convert to Judaism and well, Rape, Pillage and Convert everything in their path in classical Old Testament Style. Second criteria is easily Completed if a Large Portion of His Horde happens to be Ethnically Jewish.

3) Count Christians as Quasi-Jews, with only Kohanim being able to become Priests in any sect, while still being able to marry and have children.