Catalyzes the phosphorylation of the hydroxyl group of 4-methyl-5-beta-hydroxyethylthiazole (THZ). 5-(2-hydroxyethyl)-4-methylthiazole + ATP = 4-methyl-5-(2-phosphooxyethyl)-thiazole + ADP + H(+) Binds 2 magnesium ions per subunit. The first is coordinated via water, the second is coordinated to ATP but its significance is unclear. Cofactor biosynthesis; thiamine diphosphate biosynthesis; 4-methyl-5-(2-phosphoethyl)-thiazole from 5-(2-hydroxyethyl)-4-methylthiazole: step 1/1. Homotrimer. Belongs to the Thz kinase family.