You have broken npm by removing some of its dependencies.
are-we-there-yet is a dependency of npmlog which is a dependency of npm itself, and you somehow deleted it. The usual simple solution for such cases is reinstalling a package, but that doesn’t work if npm cannot operate. Fortunately, npm tarball comes prebundled with dependencies and hence installing npm from scratch is as simple as unpacking a tarball.
1) Go to the global node_modules directory (what would npm root -g print if you could run it):
$ cd /usr/local/lib/node_modules
2) Remove the broken npm directory or move it somewhere else (note that you might need to elevate permissions for this and the following steps):
$ mv npm /tmp
3) Download and unpack fresh npm from the registry (substitute version you want to install, e.g. 3.10.8):
$ curl -L registry.npmjs.com/npm/-/npm-{VERSION}.tgz | tar xz --transform="s:^package:npm:"
You can automate some of that with this install script:
$ curl -L https://www.npmjs.com/install.sh | sh
At this point npm should work again.