[MOB-11716] Add build steps to publish workflow (#517)

Author: pauljung14

.github/workflows/publish.yml

@@ -17,6 +17,8 @@ jobs:
           cache: 'yarn'
       - run: yarn install --frozen-lockfile
       - run: yarn test
+      - run: yarn build
+      - run: yarn build:node
       - uses: JS-DevTools/npm-publish@19c28f1ef146469e409470805ea4279d47c3d35c # v3.1.1
         with:
           token: ${{ secrets.NPM_TOKEN }}