There are two (2) distinct types of "mirrors". One is the garden variety rear-surfaced (reflective surface in back of glass - the kind on the wall in the bathroom) where light must pass through the glass twice (once going to the reflective surface, once returning). "Ordinary" window glass absorbs Ultra Violet light very effectively ... therefore much of it will be lost (actually, converted to heat).
The other type of mirror in a "First Surface" mirror, where the reflective surface has no glass to cover it: the light will be reflected without passing through *any* glass to absorb it. These will reflect UV - and most every other wavelength of light very efficiently.
The drawback to First Surface mirrors is their delicacy... to maintain flatness of the reflective surface, it must be *thin* - therefore, they are vulnerable to *any* slight contact. The acids from a normal fingerprint will penetrate immediately, destroying the coating. Cleaning them without damaging them is a definite problem. Before anyone starts ... yes, I know about "1/4 wave quartz overcoating" .... Expensive, isn't it ...!
Another possibility might be plastic. Acrylics are used a LOT for optical components and do not attenuate UV much (like 93% - 97% transmission) . The problem here would be availability ... I have *no* idea where to get "rear-surfaced" acrylic mirrors.