You can check the number of days by
// contains the difference in milliseconds
diffDate = (checkOutDate - checkInDate),
// contains the difference in days
diffDays = 1 + (diffDate / 1000 / 60 / 60 / 24);
// total cost for their stay
selectedRoomRate = $("#room").val();
selectedRoomTotalCost = selectedRoomRate * diffDays;
Updated 2016-11-11:
Updated snippet to return the cost.
Updated 2016-11-12:
<input list="roomList"> will bring you only suggestions while typing a value. To get the options to select a room type, you need <select> tag.
Try the following snippet:
function check() {
  var checkInDate = new Date($("#checkin").val()),
    checkOutDate = new Date($("#checkout").val()),
    diffDate = (checkOutDate - checkInDate),
    diffDays = 1 + (diffDate / 1000 / 60 / 60 / 24),
    selectedRoomRate = $("#room").val(),
    selectedRoomTotalCost;
  if (diffDate < 0) {
    alert("Check-out date must be after check-in date!");
    return;
  }
  selectedRoomTotalCost = selectedRoomRate * diffDays;
  alert('Total Cost: $' + selectedRoomTotalCost);
}
<!DOCTYPE html>
<html>
<meta charset=utf-8>
<meta name="viewport" content="width=device-width, initial-scale=1.0">
<head>
  <script src="https://code.jquery.com/jquery-3.1.0.js"></script>
  <link rel="stylesheet" href="settings.css">
  <title>Tranquility Hotels Limited</title>
  <script>
  </script>
</head>
<body>
  <div class="center">
    <form>
      <fieldset>
        <legend>Reserve Hotel Room</legend>
        First Name:
        <input type="text" name="firstname" placeholder="First Name" size="25" pattern="[A-Za-z-]+" title="Alphabet characters only." required>Last Name:
        <input type="text" name="lastname" placeholder="Last Name" size="25" pattern="[A-Za-z-]+" title="Alphabet characters only." required>
        <br>
        <br>Street Address:
        <input type="text" name="street" placeholder="1234 Awesome Street" size="30" pattern="[A-Za-z0-9 ]+" title="Alphabet characters and numbers only." required>City:
        <input type="text" name="city" placeholder="Cleveland" size="30" pattern="[A-Za-z ]+" title="Alphabet characters only." required>
        <br>
        <br>State:
        <input list="states" name="state" placeholder="OH" size="10" required>
        <datalist id="states">
          <option value="AL">Alabama</option>
          <option value="AK">Alaska</option>
          <option value="AZ">Arizona</option>
          <option value="AR">Arkansas</option>
          <option value="CA">California</option>
          <option value="CO">Colorado</option>
          <option value="CT">Connecticut</option>
          <option value="DE">Delaware</option>
          <option value="FL">Florida</option>
          <option value="GA">Georgia</option>
          <option value="HI">Hawaii</option>
          <option value="ID">Idaho</option>
          <option value="IL">Illinois</option>
          <option value="IN">Indiana</option>
          <option value="IA">Iowa</option>
          <option value="KS">Kansas</option>
          <option value="KY">Kentucky</option>
          <option value="LA">Louisiana</option>
          <option value="ME">Maine</option>
          <option value="MD">Maryland</option>
          <option value="MA">Massachusetts</option>
          <option value="MI">Michigan</option>
          <option value="MN">Minnesota</option>
          <option value="MS">Mississippi</option>
          <option value="MO">Missouri</option>
          <option value="MT">Montana</option>
          <option value="NE">Nebraska</option>
          <option value="NV">Nevada</option>
          <option value="NH">New Hampshire</option>
          <option value="NJ">New Jersey</option>
          <option value="NM">New Mexico</option>
          <option value="NY">New York</option>
          <option value="NC">North Carolina</option>
          <option value="ND">North Dakota</option>
          <option value="OH">Ohio</option>
          <option value="OK">Oklahoma</option>
          <option value="OR">Oregon</option>
          <option value="PA">Pennsylvania</option>
          <option value="RI">Rhode Island</option>
          <option value="SC">South Carolina</option>
          <option value="SD">South Dakota</option>
          <option value="TN">Tennessee</option>
          <option value="TX">Texas</option>
          <option value="UT">Utah</option>
          <option value="VT">Vermont</option>
          <option value="VA">Virginia</option>
          <option value="WA">Washington</option>
          <option value="WV">West Virginia</option>
          <option value="WI">Wisconsin</option>
          <option value="WY">Wyoming</option>
        </datalist>
        Zip Code:
        <input type="text" name="zipcode" placeholder="12345" size="10" pattern="[0-9]+" title="Numbers only." required>
        <br>
        <br>Phone Number:
        <input type="text" name="phone" size="25" placeholder="440-123-4567" pattern="[0-9]+" title="Numbers only." required>E-mail:
        <input type="email" name="email" size="25" placeholder="email@domain.com" pattern="[a-z0-9._%+-]+@[a-z0-9.-]+\.[a-z]{2,3}$" required>
        <br>
        <br>Check-In Date:
        <input type="date" id="checkin" name="checkin" value="2015-07-05" required>Check-Out Date:
        <input type="date" id="checkout" name="checkout" value="2015-07-17" required>
        <br>
        <br>Room:
        <select id="room" name="room" required>
          <option value="50" selected>Standard ($50/night)</option>
          <option value="75">Queen ($75/night)</option>
          <option value="100">King ($100/night)</option>
          <option value="125">Business ($125/night)</option>
          <option value="150">Deluxe Suite ($150/night)</option>
        </select>
        <button type="submit" onclick="check()">Submit</button>
      </fieldset>
    </form>
  </div>
</body>
</html>