Managed to get lxde working with hacks!

Dockerfile

@@ -9,12 +9,23 @@ RUN apt-get update -y -q && \
         novnc \
         tigervnc-standalone-server
 
+RUN apt-get install -y -q python-pip
+# https://github.com/novnc/noVNC/issues/1276
+RUN /usr/bin/pip install -U websockify==0.9.0
 
-EXPOSE 8080
-ENTRYPOINT ["bash"]
-# https://serverfault.com/questions/376302/tigervnc-ssh-without-a-vnc-password/580859#580859
-# vncserver -verbose -xstartup startlxde -SecurityTypes None -geometry 1024x768
-# /usr/share/novnc/utils/launch.sh --listen 5902 --vnc localhost:5901
+ADD websocket-path-ui-js.patch /usr/share/novnc/include
+RUN cd /usr/share/novnc/include/ && \
+    patch -p0 < websocket-path-ui-js.patch
 
-# In browser go to http://localhost:8080/vnc.html?autoconnect=true
+RUN ln -s /usr/lib/websockify/rebind.so /usr/local/lib/
 
+USER jovyan
+# RUN /opt/conda/bin/pip install jupyter-server-proxy
+RUN /opt/conda/bin/pip install https://github.com/manics/jupyter-server-proxy/archive/indexpage.zip
+ADD jupyter_notebook_config.py /home/jovyan/.jupyter/
+
+# websockify --web /usr/share/novnc 5901 -- vncserver -verbose -xstartup startlxde -SecurityTypes None -geometry 1024x768 -fg :1
+
+# Both these should work:
+# http://localhost:5901/vnc.html?autoconnect=true
+# http://127.0.0.1:8888/lxde/vnc.html?path=%2Flxde%2Fwebsockify&autoconnect=true

jupyter_notebook_config.py

+c.ServerProxy.servers = {
+    'lxde': {
+        'command': [
+            '/usr/local/bin/websockify', '--web', '/usr/share/novnc', '5901', '--', 'vncserver', '-verbose', '-xstartup', 'startlxde', '-SecurityTypes', 'None', '-geometry', '1024x768', '-fg', ':1'],
+        'absolute_url': False,
+        'environment': {
+            'PATH': '/usr/local/bin:/usr/bin:/bin',
+        },
+        'port': 5901,
+        'timeout': 30,
+        'indexpage': 'vnc.html?autoconnect=true',
+    }
+}

websocket-path-ui-js.patch

+--- ui.js.bak	2019-09-26 16:55:26.816459927 +0000
++++ ui.js	2019-09-26 16:57:06.149994873 +0000
+@@ -82,7 +82,7 @@
+     UI.initSetting('shared', true);
+     UI.initSetting('view_only', false);
+     UI.initSetting('connectTimeout', 2);
+-    UI.initSetting('path', 'websockify');
++    console.log('1', UI.getSetting('path')); console.log(); UI.initSetting('path', window.location.pathname.replace(/[^/]*$/, '').substring(1) + 'websockify'); console.log('2', UI.getSetting('path'));
+     UI.initSetting('repeaterID', '');
+ 
+     UI.rfb = RFB({'target': $D('noVNC_canvas'),
+@@ -90,7 +90,7 @@
+                   'onClipboard': UI.clipReceive,
+                   'onDesktopName': UI.updateDocumentTitle});
+ 
+-    autoconnect = WebUtil.getQueryVar('autoconnect', false);
++    autoconnect = WebUtil.getQueryVar('autoconnect', true);
+     if (autoconnect === 'true' || autoconnect == '1') {
+         autoconnect = true;
+         UI.connect();