You dont need a new engine...youre gonna have to get it off the very long and hard way....How far down into the armature is the bolt broken off?...If its like 1" you might have enough threads in the armature to thread the larger fine thread bolt into and use the broken bolt to press the armature off the shaft............
If the bolt is broken flush or where you cannot thread the fine thread bolt into it youre gonna have to use a 3 prong puller and a piece of screw to remove the armature......after that you will need a short pawl bolt and you can just bolt the pawl direct to the shaft and bolt the pullstarter with shims direct to the engine and do away with the e-starter and troublesome electrical accessories althogether.....
heres a generalization of what I mean by pressed-on in the vid below...the kid took his apart with a 3 jaw puller on a 36cc e-start engine...The e-starter systems are generally the same ...the vid sux but you can get a general idea on what it looks like...Whats funny is he didnt even realize the pullstarter works in place of the e starter body with a thin aluminum shim to space it properly......
YouTube - Youtube request: Removing e-start off of a 36cc engine