Thiazole kinase involved in thiamine salvage pathway. 5-(2-hydroxyethyl)-4-methylthiazole + ATP = 4-methyl-5-(2-phosphooxyethyl)-thiazole + ADP + H(+) kcat is 0.068 sec(-1) for thiazole. kcat is 0.069 sec(-1) for ATP. Cofactor biosynthesis; thiamine diphosphate biosynthesis; 4-methyl-5-(2-phosphoethyl)-thiazole from 5-(2-hydroxyethyl)-4-methylthiazole: step 1/1. Not regulated by thiamine or 4-methyl-5-(2-phosphonooxyethyl)thiazole. No visible phenotype. Belongs to the Thz kinase family.