How to add second icon in table header for popover menus?

[Graphictank]

Having some issues with attempting to include a filter dropdown icon to the v-date-table header. I've created the popover menu, I'm just at a loss on how to get the icon & menu incorporated into a template just like shown in the picture below. The client has stressed that they want various filters for each column (just like their excel spreadsheets have).

Here's the generic styling code I've made for a clickable icon & search popover menu:

fa-filter fa-1

                <v-card>
                  <v-list-item>
                      <v-list-item-content>
                    <v-text-field v-model="message1" label="Enter Number" dense></v-text-field>
                      </v-list-item-content>
                  </v-list-item>

                    
                      <v-card-actions>
                        <v-spacer></v-spacer>

                        <button type="button" class="btn btn-default">
                            <i class="fa fa-times"></i>&nbsp;Clear
                        </button>
                        <button type="button" class="btn btn-primary">
                            <i class="fa fa-floppy-o"></i>&nbsp;Apply
                        </button>
                      </v-card-actions>
                </v-card>
              </v-menu>

[Graphictank]

I've been able to make some updates to get it close to what I need, however the sorting arrows are now gone.

<v-data-table
@click:row="rowClick"
:headers="headers"
:items="plateList"
:search="search"
:items-per-page="10"
class="outlined elevation-0"
hide-default-header
sortable="true"
>

          <template v-slot:header="{ props: { headers } }">
          <thead>
            <tr>
              <th v-for="h in headers" v-bind:key="h">
              <v-menu attach="#menuAnchor" v-model="menu" :close-on-content-click="false" :nudge-height="200" offset-y style="display: inline-block;">
                <template v-slot:activator="{ on }" id="menuAnchor" >
                  <span v-on="on">{{h.text}}</span>
                  <v-btn :ripple="false" x-small icon v-bind="attrs" v-on="on">
                    <v-icon color="grey darken-1">fa-filter fa-1</v-icon>
                  </v-btn>
                </template>
                <v-card>
                  <v-list-item>
                      <v-list-item-content>
                    <v-text-field v-model="message1" label="Enter Plate #" dense></v-text-field>
                      </v-list-item-content>
                  </v-list-item>
                    <v-card-actions>
                        <v-spacer></v-spacer>

                        <button type="button" class="btn btn-default">
                            <i class="fa fa-times"></i>&nbsp;Clear
                        </button>
                        <button type="button" class="btn btn-primary">
                            <i class="fa fa-floppy-o"></i>&nbsp;Apply
                        </button>
                      </v-card-actions>
                </v-card>
              </v-menu>
             
              </th>
            </tr>
          </thead>
        </template> 

          </v-data-table>