You have all of the address pins constantly held low giving the address 0000, you have all of the data bits held low except for bit 1 giving a nibble (or word in the datasheet) of 0001. Using !ME and !WE (to replace the overscore notatation used for active low in the datasheet) - you have !ME always low, and (hopefully) can toggle !WE low by closing the switch - it might be worth pulling this high with a resistor when you're not trying to write, but that might be irrelevant - at the moment you've got two possible control states: !ME Lo !WE Hi, and !ME Lo and !We Lo, the truth table on the data sheet says that if both are low (the state when you have the switch closed), all data outputs will be Hi. If !ME alone is lo and !We is Hi, then you'll be in read rather than write mode, and the four outputs will show the COMPLIMENT of the inputs for the nibble you're addressing (since you've got a fixed address and a fixed input of 0001, you should see a fixed output of 1110) - so you shouldn't really see any change between holding the button or not except in the diode attached to S1 - taking !ME high should leave outputs high.
When you say LED1, do you mean LED 1 in your schematic (attached to S4) or do you mean D2 (attached to S1)? Ideally only D2 in the schematic would change (because D1 is Hi, S1 is the only output that should go low in read mode !ME lo !We Hi), if you're in write mode (both controls !ME and !WE lo) then all outputs should be high, as they should be if you take !ME high regardless of the !WE state - and if they're high they shouldn't be able to light the LEDs