Since you have started, Tom, Wood Bleach is sold at home centers under that name and is actually oxalic acid. Stains are made of different combinations of dye and pigment. The household bleach will remove the dye color and the oxalic acid will remove pigment color. You may have to use one of these, the other, or both. Follow the instructions on the product labels exactly.
It sounds to me like you have residue from the bleach left on the wood and that should be neutralized. Wiping the white look areas with a mix of 5 parts water to 1 part vinegar will do the neutralizing of the houshold bleach. Follow with a wipe down with a damp sponge with plain water.
Be careful and use eye protection and gloves with these products. And please let us know how this comes out.