Ok, firstly there's a few extra concepts you need to understand to get your circuit right.
Firstly it's that resistor you have pointed out as "Do I need this?". The answer is most certainly "NO!". You say you understand Ohm's law. If that is so, what is that resistor doing?
Yes, it is dropping the voltage. But, by how much? Well, according to Ohm's law, that is dependant on the current that's flowing through it. But what is the current flowing through it? Well, that is constantly changing, depending on which LEDs are lit, what gates are HIGH, which are LOW, etc. It's not a value you can predict, only model for different situations. So, the voltage dropped across it will be changing all the time too.
So definitely get rid of that resistor.
You also need to understand the concept of a logic input and how to wire up a switch to properly activate a logic input.
A logic input is HIGH when the voltage is above a certain threshold, and LOW when it is below a certain threshold. No voltage (i.e., not actually connected to anything) is not a valid input, and is known as a "floating" input as it is neither HIGH nor LOW. These are bad, especially if that input has an effect on the outputs you are using.
So to wire up a button so that it generates a valid logic signal it must only ever be generating one of two voltages - normally a voltage below the LOW threshold, or when pressed above the HIGH threshold. So you'd think that connect the input pin to ground for the normal LOW, and connect the button to V+ for the HIGH, so that when you press the button the input is connected to V+. Wrong. It's also connected to ground at the same time, and all you have done is connect V+ to ground and the whole circuit dies (or the battery explodes if you're unlucky). So what do you do?
This is where you need to learn about "pull down" (or pull up for active-low logic, but that's basically backwards) resistors. These are higher value resistors (typically around 10KΩ) which are used to "pull" the input towards the desired default level (ground in this case), so that when you press the button you're not creating a dead short, but just placing a large resistor across the power supply, which has little to no effect on the circuit, other than to connect your input pin properly to V+. Each button must have its own pull-down (or pull-up) resistor - they cannot be shared between buttons, or the effect is to have all the buttons do the same job at the same time. Messy.
So you will need to go shopping for some more resistors - 10KΩ is a very very common value, and you will use them lots I can promise you.
You should also learn to use proper schematic or logic capture programs. There are many free ones on the internet (some better than others). It is almost impossible to make out what your circuit is meant to do from the drawing, but a properly drawn circuit in a schematic program is so much easier to understand. Also learn the right symbols, and what they mean.
A simulator can also be a real godsend when working with discrete logic. One of the simplest and easiest I know is called Falstad and is a Java application on the web (or downloadable if you prefer) and lets you model simple circuits and check they will do what you hope they will.
Best Answer
The top part may not be connected to the silicon, it does have a small bit of copper that could be however. I'd probably give it a shot.
It really comes down to what your willing to spend in time and effort. If you don't mind resoldering the chip back to the board to see if it works, then give it a shot. If it doesn't work, then you'll have to order a new ta8229k and unsolder the old one and install the new one that has a high probability of working.