How to prove that $L_1=1^*\cup \{0^i 1^{j^2}|i\ge 0,j\ge 0\} $ is irregular?
You don't need to use the pumping lemma to prove that $L_1$ is irregular. You can also make use of the closure properties of the regular languages.

We know that regular languages are closed under intersection. That is, if $L_1$ and $L_2$ are regular languages then $L_3 = L_1\cap L_2$ is also a regular language. 

A nice way to prove that a language is irregular is to assume that it is regular and reach a contradiction using the closure properties of the regular languages. 

For example consider the language $L_p = \{1^p: p\text{ is prime}\}$ we know that this is an irregular language. Let's consider $L_q = L_p \cup 0^*$. How can we prove that this language is irregular?

Let us assume for the purpose of contradiction that $L_q$ is regular, hence $L_q\cap 1^*$ is also a regular language since $1^*$ is a regular language. But wait.. The intersection is $L_p$ (the primes) which is not a regular language. Contradiction. So $L_q$ is irregular. 

Your problem can be solved along the same lines.