Assuming you stick with the Nexus 8 and simply get the mechanical issue sorted, your total gear range is pretty well constrained. The lower gear ratio provided by that 22 tooth cog may be helpful to get up hills with a loaded touring bike, but it does take your chain gearing below the 2-2.2 range Shimano recommends for those 8 speed hubs. There's probably some engineering safety factor in that recommendation, but it's worth considering whether you might be getting up to that limit, considering your intended use of touring in the alps.
If you can mount a sprung chain tensioner (like the Shimano CT-S500, or a 'dummy' rear derailleur) to take up the chain slack, it may be possible to use a front derailleur for more top-end gear range, although it doesn't look like your frame has a derailleur hanger. an older, claw-style rear derailleur, or an emergency hanger like this might be a workable solution, but the non-turn washer presents an obstacle. If you go that route, you'll need a narrower, multi-speed chain to match the crankset and FD, and a narrower cog like the CS-S500 (2mm, available in 18 or 20 teeth). Mounting a front derailleur and multiple chainrings on a frame that wasn't designed for it, is its own messy can of worms, and a conversation you may want to have with a trusted bike shop. Other than the 2:1 minimum recommended gear ratio, issues there include BB shell width (yours appears to have a 68mm threaded bottom bracket, so that's not a limiting factor in this case), chainring size/chainstay clearance, and FD cable routing.
Edit: I based these gearing/speed estimates on what appears to be the current model year of that bike, with 700c wheels, not 26,” so my gear estimates are wrong and speeds are overstated. On the gearing front, there's also the possibility of changing to a different internally geared hub altogether, but that's a lot of expense if you don't really need a wider gear range. Even with those 38/22 tooth cogs, and assuming the Schwalbe 47mm tires specced on that bike actually measure closer to 44, the top gear on that bike is a little over 37 km/h or 23 MPH at a comfortable 100 RPM cadence. That's faster than most people ride on level ground, outside of a paceline, and pretty fast for a touring rider with panniers. when riding down a hill, you're not losing much by not having a high enough gear to keep pedaling, but you can always work on raising your cadence if you feel you're missing out just a little too much.
Most of the time spent on a bike, the limiting factor isn't gear range. Aerodynamics and comfort are key.
You may want to lower your handlebars, so that you present a smaller profile to the wind, if you can do so comfortably. Your saddle should be suitable for the angle of your back. as you lean forward, a wide saddle can hinder your leg motion and not give you the right support in the right places. a very upright riding position can also prevent you from using your more powerful leg muscles. I like my saddles to have very minimal padding, or even none at all, but it all depends on your riding style and flexibility.
In a similar vein, clip-on triathlon/time trial bars, and/or the many touring handlebars with additional hand positions, can allow you to get lower and smaller to the wind, and the movement and additional positions are especially nice on longer rides.
Form-fitting cycling clothes are great for both comfort and aerodynamics. Social comfort is a separate issue. It doesn't have to be all or nothing- there are steps between baggy and skin-tight.
I find it very helpful to ride with a cadence sensor. If I get bogged down on a hill without shifting to a low enough gear, it's much harder to recover after that hard and slow pedaling, and my speed on flat ground is more sluggish afterwards.