This took us through to the Louvre, which we had both been to before and had no real desire to visit again, being so large and somewhat overwhealming. We did, however, pop into the tourist office there which managed to provide us with almost zero information but did have an internet kiosk so we quickly checked up on some facts. We also visited the boutique chocolate shop, Salon du Chocolat, and bought some amazingly rich eclairs, which sufficiently boosted our blood-sugar levels to motivate us to head back out into the cold.
We wandered into the Jardin du Palais Royal nearby, since neither of us had been there before, and were impressed by the (sometimes silly) sculpture on display. We strolled through the gardens and around the mostly-closed shops around the perifery of the courtyard before heading out again.
To complete the day of sight-seeing we walked down to the Île de la Cité where Notre Dame is located. We queued up outside the cathedral (apparently post-Christmas is a busy time in Paris) and walked through the inside, which is really quite impressive.
By this time it was really getting quite dark, which is when I realised just how amazing the city looks at night. While during the day it is mostly shades of grey (especially given the overcast sky), at night the monuments and bridges are all up-lit and the sky takes on the colours of the city lights. I don't think we saw a single star the whole time we were in Paris.
We had dinner in a crêperie that Linda had discovered on her previous trip to Paris (and a place we would go back to several more times during out stay, due to its good quality food that was priced right for our budget), and spent the remainder of the day wandering around the islands on the Seine and ultimately climbing up the Arc de Triomphe to check out the view of the city at night.