In android playing videos from web-view is such a mess but now just using a small trick you can play all web videos of the YouTube, Daily-motions, etc. You just have to add this method in your web-view Activity.

Here is the trick-:

          mWebView.setWebChromeClient(new  WebChromeClient() {
        public CustomViewCallback mCustomViewCallback;

        @Override
        public void onShowCustomView(View view, CustomViewCallback callback) {
            super.onShowCustomView(view, callback);
            if (view instanceof FrameLayout) {
                FrameLayout customViewContainer = (FrameLayout) view;
                mCustomViewCallback = callback;
                if (customViewContainer.getFocusedChild() instanceof VideoView) {
                    VideoView customVideoView = (VideoView) customViewContainer.getFocusedChild();
                    try {
                        Field mUriField = VideoView.class.getDeclaredField("mUri");
                        mUriField.setAccessible(true);
                        Uri uri = (Uri) mUriField.get(customVideoView);
                        Log.w("uri", ""+uri);
                        Intent intent = new Intent(Intent.ACTION_VIEW);
                        intent.setDataAndType(uri, "video/*");
                        startActivity(intent);
                        new Handler().post(new Runnable() {
                            @Override
                            public void run() {
                                mCustomViewCallback.onCustomViewHidden();
                            }
                        });
                    } catch (Exception e) {
                    }
                }
            }
        }
             });



Make sure you add **webview.getSettings().setPluginsEnabled(true);** 
to your webview settings.
And most importantly, for any webpage to load in a WebView, be sure to have the INTERNET permission in your Manifest:

**<uses-permission android:name="android.permission.INTERNET"/>**

I hope this will help CHEERS