Mueller Hut is in the Mt Cook National Park, which is in New Zealand's South Island.
The trek to Mueller Hut is only 4.6km, but it is all up hill, so allow 4 hours or so. The trek starts from Mount Cook Village, or if you're staying at the White Horse Campsite you can start from there (it's slightly closer).
On a nice day, the scenery is absolutely breath taking. I imagine it would still be pretty amazing on a not so nice day too. Throughout the whole walk, you have beautiful views to Mt Cook (New Zealand's highest mountain) and across Mt Cook Village. At one point you walk over a ridge to see Mt Sefton (New Zealand's 3rd highest mountain) right in front of you - what a spectacular sight!
Mueller Hut itself looks across at both Mt Cook and Mt Sefton. You can even sit on the porch and watch (and hear) snow avalanches crashing down Mt Sefton.