Use a cronjob, but not to run your pods, but to schedule a Kubernetes API command that will restart the deployment everyday (

kubectl rollout restart

Rollouts create new ReplicaSets, and wait for them to be up, before killing off old pods, and rerouting the traffic. Service will continue uninterrupted.

You have to setup RBAC, so that the Kubernetes client running from inside the cluster has permissions to do needed calls to the Kubernetes API.

---
# Service account the client will use to reset the deployment,
# by default the pods running inside the cluster can do no such things.
kind: ServiceAccount
apiVersion: v1
metadata:
  name: deployment-restart
  namespace: <YOUR NAMESPACE>
---
# allow getting status and patching only the one deployment you want
# to restart
apiVersion: rbac.authorization.k8s.io/v1
kind: Role
metadata:
  name: deployment-restart
  namespace: <YOUR NAMESPACE>
rules:
  - apiGroups: ["apps", "extensions"]
    resources: ["deployments"]
    resourceNames: ["<YOUR DEPLOYMENT NAME>"]
    verbs: ["get", "patch", "list", "watch"] # "list" and "watch" are only needed
                                             # if you want to use `rollout status`
---
# bind the role to the service account
apiVersion: rbac.authorization.k8s.io/v1
kind: RoleBinding
metadata:
  name: deployment-restart
  namespace: <YOUR NAMESPACE>
roleRef:
  apiGroup: rbac.authorization.k8s.io
  kind: Role
  name: deployment-restart
subjects:
  - kind: ServiceAccount
    name: deployment-restart
    namespace: <YOUR NAMESPACE>

And the cronjob specification itself:

apiVersion: batch/v1
kind: CronJob
metadata:
  name: deployment-restart
  namespace: <YOUR NAMESPACE>
spec:
  concurrencyPolicy: Forbid
  schedule: '0 8 * * *' # cron spec of time, here, 8 o'clock
  jobTemplate:
    spec:
      backoffLimit: 2 # this has very low chance of failing, as all this does
                      # is prompt kubernetes to schedule new replica set for
                      # the deployment
      activeDeadlineSeconds: 600 # timeout, makes most sense with 
                                 # "waiting for rollout" variant specified below
      template:
        spec:
          serviceAccountName: deployment-restart # name of the service
                                                 # account configured above
          restartPolicy: Never
          containers:
            - name: kubectl
              image: bitnami/kubectl # probably any kubectl image will do,
                                     # optionaly specify version, but this
                                     # should not be necessary, as long the
                                     # version of kubectl is new enough to
                                     # have `rollout restart`
              command:
                - 'kubectl'
                - 'rollout'
                - 'restart'
                - 'deployment/<YOUR DEPLOYMENT NAME>'

Optionally, if you want the cronjob to wait for the deployment to roll out, change the cronjob command to:

command:
 - bash
 - -c
 - >-
   kubectl rollout restart deployment/<YOUR DEPLOYMENT NAME> &&
   kubectl rollout status deployment/<YOUR DEPLOYMENT NAME>