Check if object contains only empty nested objects
Some issues?


typeof is not a function. eg typeof(input) === 'object' is written typeof input === 'object'
null is an Object so when you test for an object using typeof you MUST make sure it is not null as well.
The test that you comment as // catch null or undefined object will catch any value that evaluates to a falsey, such as false, 0, ""


What is empty?

As it is unclear what you define as empty so I will make some assumptions, they are arbitrary and may or may not fit your needs.


An array is empty if it contains only empty items.
An object is empty if it is null or contains no own properties. If it has properties that are null or (defined as) undefined it is not empty.
A string is empty if it contains no characters
Empty items are null, undefined, {}, [], and ""


Examples

Empty

[], 
{}, 
[undefined],
[null],
[[], [], []], 
[{},,[],[[[null],[undefined]],[,,,,,]],
new Array(10),
null,
undefined,
""


Not empty

[0]
[{A:null}], 
{A:undefined}, 
[,,,0],
[[], [], [1]], 
[{},,[],[[[1],[]],[]],
(new Array(10))[1] = 0,
false,
true,
"  ",


Rewrite

With the above assumptions you can rewrite the code as a two functions.

As a non empty object mean we return false, and thus we do not need to iterate its values.

The entry point is isItemEmpty you would call it as you did isDeepEmpty 

const isObjEmpty = obj =&gt; obj === null || Object.keys(obj).length === 0;
const isItemEmpty = item =&gt; item === undefined || item === "" ||
    (Array.isArray(item) &amp;&amp; array.every(isItemEmpty)) ||
    (typeof item === "object" &amp;&amp; isObjEmpty(item));


Usage

isItemEmpty([{},[],[[]]]);    // returns true
isItemEmpty([{A:0},[],[[]]]); // returns false
isItemEmpty("");              // returns true