MD simulations of the peptide Trp-cage dissolved in a solvent composed of 28% 1,1,1,3,3,3-hexafluoroisopropanol-water and contained within a nanosphere of 4.2 nm radius are described. To provide a thermal buffer, the nanosphere is submerged in a collection of liquid neopentane molecules, modeled as united atoms. It was found that the HFIP-water mixture demixes when confined under these conditions, with most of the fluoroalcohol becoming strongly associated with the walls of the nanosphere. The remaining HFIP interacts with the peptide and itself in the interior of the nanoshell. Diffusion of HFIP molecules near the surface of the nanoshell is limited, taking place over a small volume, while diffusion of the fluoroalcohol in the center of the nanoshell is more wide-ranging. By contrast, diffusion of the water in the solvent mixture appears to take place equivalently throughout the shell. The conformation of the peptide, distribution of solvent components around it, and the number and duration of the contacts with solvent molecules are calculated and contrasted to results previously reported for the bulk (non-nanocontained) system.