What did I do wrong with this switching voltage regulator circuit (LM2574)?
You have committed two errors:

First: The data sheet shows a direct connection from the supply output voltage to the feedback.  You put a resistor in series.  The data sheet hints that the feedback pin draws current -- there are resistors in there.  That means that the feedback pin will see a lower voltage than the output, and the only thing the chip "knows" is to regulate the feedback to 5V.  So it is natural to expect that putting a resistor in series will get you a higher output voltage.

Second: LED's, or at least bare LEDs, are junction diodes.  This means that their current increases exponentially with increasing voltage.  Diodes vary, and LEDs more so, but the current will double for every 10 or 20mV of increase in voltage.  This means that over the range of currents where you can see light and the LED isn't burning up, the LED will "try" to be a constant-voltage load.  

There are hobby projects out there that connect LEDs up straight to dry cells.  This works because a dry cell (particularly a cheap one) has some internal resistance, and a voltage that'll naturally light up a red LED.

However, if you connect an LED to a stiff voltage source you'll most likely get no light, or you'll burn up the LED.  For reliable operation you must put a current-limiting resistor in series with it.

So, congratulations, you've made an LED burner-upper.

Get rid of the resistor in the feedback path.  You can't reliably light two of your LEDs in series off of 5V without fancy circuitry, so wire each one in parallel with a suitable resistor.  "Suitable", in your case, is something that puts 30mA through the LED when it is dropping 1.8V: $$R = \mathrm{\frac{5V - 1.8V}{30mA}} = 106\Omega$$  Round that up to 110 ohms and you'll get plenty of bright light.