First up, no, you should not expect that the SR will read your button press.
The VIL for the 74HC family of chips is 0.8V.
The VIH for these is 2.0V.
The WHAT?
VIL = The maximum voltage
allowed as a ‘low’ (logic 0) at an input.
VIH = The minimum voltage allowed as a ‘high’ (logic 1) at an input.
This means that a switch intended to pull the input low, must present to that input a voltage of less than 0.8V. Anything higher than this and results are not guaranteed.
1.6V falls right into the undetermined place between VIL and VIH
The 10k resistors are a pull-up device, intended to give the PIC / SR chips a steady high state (logic 1) when the button is not pressed. Changing these for LEDs with resistors might lead to the SR reading incorrectly.
Now, you may find that results are better if you keep what you have, but connect the SR pin to the junction of button+LED, not the LED+220R.
ie:
[tt]Vdd—220R—LED—±–BUTTON—VssGND
|
To DIN SR pin[/tt]
Try that, take some measurements…
If it doesn’t look good, how about this?
[tt]
Vdd-------10k-------+
|
Vdd----220R----LED–±—button/SR pin-----Vss(ground)
[/tt]
P.S.
** Edit: the 1.6V you are reading is the voltage drop across your LED **
(Don’t forget to answer the question…) 
Edit:typo