Listing DaskGateway clusters created via Python code alongside those created via the dask labextension UI

**What happened**:

I can create a dask-gateway cluster via the dask-labextension view and I'll see it visible there then.


But, if I create a dask-gateway cluster from a notebook using code like below, then no dask cluster shows up in the list of clusters.

```python
from dask_gateway import Gateway
gateway = Gateway()
cluster = gateway.new_cluster()
```

**My wish**

My wish is that the dask clusters I've created should be listed visually. I'm not sure if this is possible or not, but I'd like to describe this wish here to explore if we can make it happen one way or another.

**Environment**:

JupyterHub (1.1.1 Helm chart) + Dask-Gateway (0.9.0 Helm chart).

```
$ conda list | grep dask
dask 2021.6.0 pyhd8ed1ab_0 conda-forge
dask-core 2021.6.0 pyhd8ed1ab_0 conda-forge
dask-gateway 0.9.0 py38h578d9bd_0 conda-forge
dask-glm 0.2.0 py_1 conda-forge
dask-kubernetes 2021.3.1 pyhd8ed1ab_0 conda-forge
dask-labextension 5.0.2 pyhd8ed1ab_0 conda-forge
dask-ml 1.9.0 pyhd8ed1ab_0 conda-forge
pangeo-dask 2021.06.05 hd8ed1ab_0 conda-forge
```

```
$ python --version
Python 3.8.10
```

Operating System: Ubuntu 20.04
Install method: conda-forge

```
# The current environment and dask configuration via environment
DASK_DISTRIBUTED__DASHBOARD_LINK=/user/{JUPYTERHUB_USER}/proxy/{port}/status
DASK_GATEWAY__ADDRESS=http://10.100.116.39:8000/services/dask-gateway/
DASK_GATEWAY__AUTH__TYPE=jupyterhub
DASK_GATEWAY__CLUSTER__OPTIONS__IMAGE={JUPYTER_IMAGE_SPEC}
DASK_GATEWAY__PROXY_ADDRESS=gateway://traefik-prod-dask-gateway.prod:80
DASK_GATEWAY__PUBLIC_ADDRESS=/services/dask-gateway/
DASK_LABEXTENSION__FACTORY__CLASS=GatewayCluster
DASK_LABEXTENSION__FACTORY__MODULE=dask_gateway
DASK_ROOT_CONFIG=/srv/conda/etc
```