There is a (pretty pointless in most cases ;)) workaround to keep your sphere as shiny as a mirror and still have the HDRI blurry.
You can use a ray switch material and let it send out a blurry version of your HDRI to anything reflective.
But, if in any case, you want everything blurry in the sphere, really set the blurriness in the material and not in each and every scene material via ray switcher ;)
PS the ray switcher is NOT a pointless shader, this usecase is just a bit ... pointless, unless you really just want the HDRI blurry and the rest sharp.