In Dienekes K13 analysis Nganassan, Evenk and Yakut have highest, above 80%. Their autosomal DNA is probably not exactly the same as East Asians, however. They have East Asian genes, European genes, and proabably genes of their own.
9.3% for Finns seems high - most estimates are in the range of 5-6%, for example Dienekes K13. In Russian it varies between somewhat higher and significantly lower.
Sarkissian et al analysed ancient European remains in the article "Ancient DNA Reveals Prehistoric Gene-Flow from Siberia in the Complex Human Population History of North East Europe". They found that some ancient people who were living in Europe were clearly Siberian, but still not related to the Siberian admixture in Saamis! So, as the title suggest, Eastern Europe has a complex population history, with both caucasoids and mongoloids turning up far from were we are used to see them today.
As for Finns, the most widely held theory is that the proto-Saami speaking people, moving from the Uralic Urheimat in the Volga region, picked up Siberian genes from a population in North-Eastern Europe. They then moved to Lapland and mixed with the paleo-European population to form present day Saamis. Finns in turn got their Siberian admixture from the proto-Saami speaking people.
This explain why all Uralic people do not have Siberian genes. For example Veps lack it, and it's not found in the Baltic although Finnic languages were previously spoken there.