WPF AttachedProperty to show the on-screen keyboard when the control get focused by touch then hide it when the control loses focus. Code in this Gist is MIT Licensed.

MainWindow.mxml

<Window x:Class="TouchKeyboardTest.MainWindow"
		xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation"
		xmlns:x="http://schemas.microsoft.com/winfx/2006/xaml"
		xmlns:local="clr-namespace:TouchKeyboardTest"

		xmlns:d="http://schemas.microsoft.com/expression/blend/2008"
		xmlns:mc="http://schemas.openxmlformats.org/markup-compatibility/2006"
		mc:Ignorable="d"

		Title="Touch Keyboard Test"
		Height="350"
		Width="525">

	<Grid Margin="25">
		<StackPanel>
			<TextBox Margin="0 5" local:TabletHelper.ShowInputPanelOnTouch="True" />
			<DatePicker Margin="0 5" local:TabletHelper.ShowInputPanelOnTouch="True" />
			<RichTextBox Margin="0 5" local:TabletHelper.ShowInputPanelOnTouch="True" />
			<Button Content="Clear Focus" HorizontalAlignment="Right" />
		</StackPanel>
	</Grid>
</Window>

TabletHelper.cs

using System;
using System.Diagnostics;
using System.IO;
using System.Runtime.InteropServices;
using System.Threading;
using System.Threading.Tasks;
using System.Windows;
using System.Windows.Input;

namespace TouchKeyboardTest
{
	public class TabletHelper
	{
		public static readonly DependencyProperty ShowInputPanelOnTouchProperty =
			DependencyProperty.RegisterAttached("ShowInputPanelOnTouch", typeof(bool), typeof(TabletHelper), new FrameworkPropertyMetadata(false, ShowInputPanelOnTouchChanged));

		[DllImport("user32.dll")]
		public static extern int FindWindow(string lpClassName, string lpWindowName);

		[DllImport("user32.dll")]
		public static extern int SendMessage(int hWnd, uint Msg, int wParam, int lParam);

		public const int WM_SYSCOMMAND = 0x0112;
		public const int SC_CLOSE = 0xF060;

		private const string TabTipPath = @"microsoft shared\ink\tabtip.exe";
		private const string OskPath = @"osk.exe";

		readonly static string _expandedTabTipPath;
		readonly static string _expandedOskPath;

		static CancellationTokenSource _cancelCloseInputPanel;

		static TabletHelper()
		{
			var programFilesPath = Environment.GetFolderPath(Environment.SpecialFolder.CommonProgramFiles).Replace(" (x86)", "");
			_expandedTabTipPath = Path.Combine(programFilesPath, TabTipPath);

			var system32Path = Environment.GetFolderPath(Environment.SpecialFolder.System);
			_expandedOskPath = Path.Combine(system32Path, OskPath);
		}

		public static void ShowInputPanel()
		{
			if (_cancelCloseInputPanel != null)
				_cancelCloseInputPanel.Cancel();

			if (File.Exists(_expandedTabTipPath))
				Process.Start(_expandedTabTipPath);
			else if (File.Exists(_expandedOskPath))
				Process.Start(_expandedOskPath);
		}

		public static void HideInputPanel()
		{
			_cancelCloseInputPanel = new CancellationTokenSource();
			Task.Delay(100, _cancelCloseInputPanel.Token)
				.ContinueWith(t =>
				{
					_cancelCloseInputPanel = null;

					if (t.IsCanceled)
						return;

					CloseInputPanelWindow();
				});
		}

		private static void CloseInputPanelWindow()
		{
			int inputPanelHandle = FindWindow("OSKMainClass", null);
			if (inputPanelHandle <= 0)
				inputPanelHandle = FindWindow("IPTIP_Main_Window", null);

			if (inputPanelHandle > 0)
				SendMessage(inputPanelHandle, WM_SYSCOMMAND, SC_CLOSE, 0);
		}

		public static bool GetShowInputPanelOnTouch(DependencyObject obj)
		{
			return (bool)obj.GetValue(ShowInputPanelOnTouchProperty);
		}

		public static void SetShowInputPanelOnTouch(DependencyObject obj, bool value)
		{
			obj.SetValue(ShowInputPanelOnTouchProperty, value);
		}

		private static void ShowInputPanelOnTouchChanged(DependencyObject obj, DependencyPropertyChangedEventArgs e)
		{
			var uiElement = (UIElement)obj;
			uiElement.GotKeyboardFocus -= UIElement_GotKeyboardFocus;
			uiElement.TouchDown -= UIElement_TouchDown;

			if ((bool)e.NewValue)
			{
				uiElement.GotKeyboardFocus += UIElement_GotKeyboardFocus;
				uiElement.TouchDown += UIElement_TouchDown;
			}
		}

		private static void UIElement_TouchDown(object sender, TouchEventArgs e)
		{
			var uiElement = (UIElement)sender;
			uiElement.TouchLeave -= UIElement_TouchLeave;
			uiElement.TouchUp -= UIElement_TouchUp;

			if (!uiElement.IsFocused)
				return;

			uiElement.TouchLeave += UIElement_TouchLeave;
			uiElement.TouchUp += UIElement_TouchUp;
		}

		private static void UIElement_TouchLeave(object sender, TouchEventArgs e)
		{
			var uiElement = (UIElement)sender;
			uiElement.TouchLeave -= UIElement_TouchLeave;
			uiElement.TouchUp -= UIElement_TouchUp;
		}

		private static void UIElement_TouchUp(object sender, TouchEventArgs e)
		{
			var uiElement = (UIElement)sender;
			uiElement.TouchLeave -= UIElement_TouchLeave;
			uiElement.TouchUp -= UIElement_TouchUp;

			ShowInputPanel();
			uiElement.LostKeyboardFocus += UIElement_LostKeyboardFocus;
		}

		private static void UIElement_GotKeyboardFocus(object sender, KeyboardFocusChangedEventArgs e)
		{
			var uiElement = (UIElement)sender;
			uiElement.LostKeyboardFocus -= UIElement_LostKeyboardFocus;

			if (!uiElement.AreAnyTouchesOver)
				return;

			ShowInputPanel();
			uiElement.LostKeyboardFocus += UIElement_LostKeyboardFocus;
		}

		private static void UIElement_LostKeyboardFocus(object sender, KeyboardFocusChangedEventArgs e)
		{
			var uiElement = (UIElement)sender;
			uiElement.LostKeyboardFocus -= UIElement_LostKeyboardFocus;

			HideInputPanel();
		}
	}
}