Kiruna the most nothern town in Sweden. It is a famous iron mining town so don't be scared if you hear explosion at night that shake whatever building you might be staying in. It also has the famous ice hotel and, most importantly, one of the best views of Aurora Borealis.
In Kiruna there are only two main roads heading in the direction you want to go. These streets are Lombololeden or Malmvägen, it is a small town so if you head down either of these roads to the East then you'll eventually find the roundabout where the two of them meet. Only Lombololeden follows the E10 the entire way.
Smaller villages around Kiruna (Svappavaara, Vittangi, Gällivare...) offer significantly lower chances of getting a ride than from Kiruna. One should consider rejecting a ride to a halfway town and waiting for a better ride instead. On the other hand, these towns have small hospitals that are open (but not running) 24h, in case one gets stranded there.