pEXP23

I have a few comments to what you said until now:
cloning part - you are not sure whether the cloning worked properly i.e. several bands after digestion; I always sequence what I clone before I throw it in the expression strain; I check and double check that everything is in frame with the tag, start codon and stop codon
expression part - it looks to me that you have degradation products of your protein; what happens is that when bacteria fail to fold your protein properly, the internal machinery of the cell degrades the misfolded protein (an alternative pathway to inclusion bodies) leading to you having your protein and some other smaller degradation products. To be sure that those are degradation products, besides WB, I usually do a small enrichment experiment of cell lysate from 50 mL of expression resuspended in 5 mL buffer followed by sonication; incubate with 20 uL of Ni-NTA beads (2 h in cold room on rotating wheel), wash 3-5 times with buffer + 30 mM imidazole at lowest centrifugation speed; boil the collected beads after the wash in SDS loading buffer and run on SDS-PAGE); this fast experiment has the purpose of showing you whether you can get rid of the degradation products through affinity purification aka: do the degradation products also have the His-tag or are from the opposite terminus? also you get the answer to the question: are they degradation products of your protein? and is the His-tagged protein correctly expressed with the His-tag accessible for purification? (this is my expression test, besides the classical band of the expected size after induction) Anyhow this would not solve your problem, just makes it easier to characterize

Your protein is ~17kDa. I would not expect it to be difficult to express, but I also had such a small protein but tricky. The solution was to reclone it in pGex to have it with a GST tag instead of His-tag. More, pGex has a milder promoter. Sometimes the T7 promoter is too strong for some proteins and produces too many proteins that do not have time to fold and they rather aggregate (there are several ribosomes on the same mRNA and in bacteria the peptide is first translated completely before it starts to fold, so emerging peptides from 2 close ribosomes would aggregate faster if too many mRNAs are present) pGex vector has a tac promoter, way milder than T7. Alternatively, if you have enough of your protein at full length and you just want to get rid of the degradation products which, for under 17 kDa I do not think it is possible through gel filtration, you can reclone your protein in the same vector with a N-terminal Strep tag (or any other affinity tag) and a C-terminal His tag. If you purify your protein in two affinity steps, you will get only the protein that has both tags aka it is full length. Bonus: your protein will be clean-clean

If all this recloning sounds like too much work, then you can (and actually this is what you should try first) optimize your expression conditions. Try different concentrations of IPTG: 0.1, 0.5 and 1 mM are the ones I try. Also different temperatures. I try usually 18, 22, 25 (3h induction and ON) and 30 and 37oC (2h and 4h induction). And yes, 18, 22 and 25 oC give different results even though they are close together. I have a protein that is optimally expressed at 25 while one of its domains is optimally expressed at 22 (makes you think about expressing ON at room temperature, doesn't it? )

Codon optimization comment: Yes, the rumor has it that if your gene has rare codons, you might get fragments expressed. However, I have read quite a bit on the matter and there is also another school of thought (to which I adhere). The thing is that rare does not mean inexisting. What happens (or we imagine to happen) on the molecular level (and we have no way of knowing exactly) is that when the ribosome needs one of the rare t-RNAs, it pauses on that position until he gets the needed t-RNA. It can fall when it waits too long and give out the smaller protein, but it can also just wait and lead to smaller yields in the protein (which is the theory I believe in). It can even be that it will incorporate the wrong amino acid. We have no way of knowing what really happens. The truth is that it leads to less expression of your protein. To be sure that you are not in this situation, I recommend this tool: http://gcua.schoedl.de/ But, I have seen wonderful expression (20-30 mg/L) even when I had 4-5 rare codons in my protein (hence my disbelief in rare codons theory) But for every exception from the rare codons (like my case) there are 100 of cases in which checking the rare codons and changing the strain was the solution. I just wanted to comment that is not the universal solution.

In fact, with expression is a series of trial and errors. Only like 30% of proteins (or smth like this I have seen once in a review) under 50 kDa can be expressed in pET vectors with IPTG induction in LB media. I always say: if it is expressable in LB with IPTG, smb would have already done it and my PhD project would be too easy.