If it's dual pane window... temp drop probably won't be an issue, but sun shining in might be.
In my bathroom, the humidity and temp fluctuate quite a bit when people shower especially. You might consider putting a larger cardboard box around yours or just wherever you think is the most temperature stable location. For me it's an inside wall, the corner of my living room and not directly above or below a heater vent. Some folks have used a hall closet.
I MUST use towels (unused fluffy pajammies) on my bators to get them up to temp as well, especially the first year when I kept my house at 58. These days I've been keeping it set at 64. Summer hatching is basic no go as the windows stay open and house temp swings about 10 degrees or more daily. Both weird and great how it can be different strokes for different folks! When I did live in So Cal, would've been a different story as AC ran constantly and rarely ever heated the place but air quality was so bad windows stayed shut even on nice days... It's important NOT to cover or block your fresh air holes. I have also heard of someone overheating their "brain" and having to replace it... so I do take the precaution of uncovering that area if possible. I would probably wrap the side better... and make sure the turner is turning all the way.
The one I had like that... to many temp variations throughout the whole thing...The insulation should help and moving eggs to a new location inside daily MIGHT help. When I used mine as a double decker, made it worse. But, no problem working with what ya got and making the best of it... especially when it didn't cost ya.
Marans eggs and blue ones can be especially difficult to see into when candling. Doing it in a very dark space can help. My cell phone flashlight works quite well actually. I hold it to the fat end and try to see the air cell mostly. Sometimes I am able to make out veins and such. Lighter eggs are so much more fun to candle... what kinda rooster ya got already? Candling is not my strong suit but I would start looking... well, I'm gonna say day 7...
more like 5 really. Anyways, practice it and have fun! You really don't have to worry TOO much about them cooling off as long as you aren't doing it excessively often or slow. Remember a mama would leave them for about 10 minutes once or twice per day depending on how harsh the weather. 
I guess the main thing is be careful not to drop them!!One of my favorite hatching resources with lots of fantastic information..
Incubation guide
Some old timers claim NOT the best or over complicated. I think it's a great starting point and as you expand your experience, understanding, and knowledge you may be able to tweak things a TAD better in your favor... I guess the whole dark egg, lower humidity is a good example, as well as don't believe everything you read!
Some of us won't help chicks hatch while others do. You've already started off showing that you WILL go against the grain and try stuff!
Hope you tag me so I can see your results. There are lot's of hatch alongs that you might join as well. I have unexpectedly sat eggs or planned a cool hatch date like new years. Getting new life is just so much fun! Happy chickeneering.
Maybe that's because I only started noticing them on day 7 or so for my broody hatched eggs.
