以下のコードを呼び出し

new L.Draw.Marker(map, drawControl.options.marker).enable();

ユーザーコントロール：ourCustomControl を作成してmapに登録。
ちなみにユーザーコントロールのクリックイベントが発生するとMAPにもイベントが伝搬してしまうので、
クリックイベントではe.stopPropagation();を読んでMAPにイベントだ伝搬することを停止。

var ourCustomControl = L.Control.extend({
	options: {
		position: 'topleft' 
		//control position - allowed: 'topleft', 'topright', 'bottomleft', 'bottomright'
	},
	onAdd: function (map) {
		var container = L.DomUtil.create('div', 'leaflet-bar leaflet-control leaflet-control-custom');
		container.style.backgroundColor = 'white';
		container.style.width = '30px';
		container.style.height = '30px';
		container.onclick = function(e){
			e.stopPropagation();
		  	console.log('buttonClicked');
			//drawControl.options.draw = true;
			new L.Draw.Marker(map, drawControl.options.marker).enable();
		}
		return container;
	},
});

map.addControl(new ourCustomControl());

参考Creating custom control button in leaflet

LeafLet DrawはLeaflet上でFeatureを登録編集するプラグイン

GITHUBのURL

https://github.com/Leaflet/Leaflet.draw

基本的な使い方
githubよりSRC以下をダウンロードしてリンク

	<script src="../lib/leafret.draw/src/Leaflet.draw.js"></script>
    <script src="../lib/leafret.draw/src/Leaflet.Draw.Event.js"></script>

    <script src="../lib/leafret.draw/src/Toolbar.js"></script>
    <script src="../lib/leafret.draw/src/Tooltip.js"></script>

    <script src="../lib/leafret.draw/src/ext/GeometryUtil.js"></script>
    <script src="../lib/leafret.draw/src/ext/LatLngUtil.js"></script>
    <script src="../lib/leafret.draw/src/ext/LineUtil.Intersect.js"></script>
    <script src="../lib/leafret.draw/src/ext/Polygon.Intersect.js"></script>
    <script src="../lib/leafret.draw/src/ext/Polyline.Intersect.js"></script>
    <script src="../lib/leafret.draw/src/ext/TouchEvents.js"></script>
    <script src="../lib/leafret.draw/src/draw/DrawToolbar.js"></script>
    <script src="../lib/leafret.draw/src/draw/handler/Draw.Feature.js"></script>
    <script src="../lib/leafret.draw/src/draw/handler/Draw.SimpleShape.js"></script>
    <script src="../lib/leafret.draw/src/draw/handler/Draw.Polyline.js"></script>
    <script src="../lib/leafret.draw/src/draw/handler/Draw.Marker.js"></script>
    <script src="../lib/leafret.draw/src/draw/handler/Draw.Circle.js"></script>
    <script src="../lib/leafret.draw/src/draw/handler/Draw.CircleMarker.js"></script>
    <script src="../lib/leafret.draw/src/draw/handler/Draw.Polygon.js"></script>
    <script src="../lib/leafret.draw/src/draw/handler/Draw.Rectangle.js"></script>
    <script src="../lib/leafret.draw/src/edit/EditToolbar.js"></script>
    <script src="../lib/leafret.draw/src/edit/handler/EditToolbar.Edit.js"></script>
    <script src="../lib/leafret.draw/src/edit/handler/EditToolbar.Delete.js"></script>
    <script src="../lib/leafret.draw/src/Control.Draw.js"></script>
    <script src="../lib/leafret.draw/src/edit/handler/Edit.Poly.js"></script>
    <script src="../lib/leafret.draw/src/edit/handler/Edit.SimpleShape.js"></script>
    <script src="../lib/leafret.draw/src/edit/handler/Edit.Rectangle.js"></script>
    <script src="../lib/leafret.draw/src/edit/handler/Edit.Marker.js"></script>
    <script src="../lib/leafret.draw/src/edit/handler/Edit.CircleMarker.js"></script>
    <script src="../lib/leafret.draw/src/edit/handler/Edit.Circle.js"></script>

    <link rel="stylesheet" href="../lib/leafret.draw/src/leaflet.draw.css"/>

以下のコードでツールバーが表示され、編集等が可能となる

var drawnItems = new L.FeatureGroup();
map.addLayer(drawnItems);

var drawControl = new L.Control.Draw({
     edit: {
         featureGroup: drawnItems
     }
 });
 map.addControl(drawControl);

map.on(L.Draw.Event.CREATED, function (event) {
	var layer = event.layer;
	drawnItems.addLayer(layer);
});

登録用ツールバーを消去

var drawControl = new L.Control.Draw({
	 draw : false,
     edit: {
		featureGroup: drawnItems
     },
 });

編集削除ツールバーも消去

var drawControl = new L.Control.Draw({
	 draw : false,
     edit: {
		featureGroup: drawnItems,
		edit: false,
		remove: false
     },
 });

消去を再表示はできない？

一旦、コントロールそのものを削除して再登録だな。

削除のコード

 map.removeControl(drawControl);