Thiazole kinase involved in thiamine salvage pathway. 5-(2-hydroxyethyl)-4-methylthiazole + ATP = 4-methyl-5-(2-phosphooxyethyl)-thiazole + ADP + H(+) kcat is 1.79 sec(-1) for thiazole. kcat is 2.39 sec(-1) for ATP. Cofactor biosynthesis; thiamine diphosphate biosynthesis; 4-methyl-5-(2-phosphoethyl)-thiazole from 5-(2-hydroxyethyl)-4-methylthiazole: step 1/1. Belongs to the Thz kinase family.