HEK 293 cells are generally less robust than E.coli, and can be disrupted quite easily. My recommendation would be to pellet the cells (at low speed to prevent cell lysis) and then resuspend the cells in some lysis buffer. Tris/Phosphate/HEPES buffers at a physiological pH are probably a good bet, with some NaCl to bulk up the ionic strength (~150mM). Protease inhibitors are pretty important, especially with HEK cells. It sort of depends on your protein, but you could use something like PMSF, or some of those protease inhibitor cocktail tablets. Given that you want to do Ni-NTA add about ~5-20mM imidazole (to prevent non-specific binding) and stay away from EDTA.
I would also recommend keeping everything on ice and, if possible, pre-chilling the lysis buffer. Lysis should be accomplished easily by a few pulses with a sonicator or through multiple freeze-thaw cycles. Experimentation will be needed to see what works best for your system.
Centrifuge at high-ish speed (~18,000g) to remove any debris before applying to the HisTrap HP column.
I would first centrifuge the cells and resuspend the pellet in pure milliQ water. It takes about 30min to disrupt the cells. Sonication would definitely help if you have the equipment. Depending on the volume, you can either do ultracentrifugation or 0.45 um syringe filtering the sample and load it to the column.
It is depends also from the amount of cells that you would like to lisate.
For small lisis (max 1g of cell wet biomass), you can use also CellLitc M https://www.sigmaaldrich.com/life-science/proteomics/recombinant-protein-expression/cell-lysis/mammalian-cell-lysis/cellytic-m.html )
just a question: Why are you performing intracellular expression and not secretion on the media? Because one of the great advantage of mammalian is the possibility to secrete the protein out of the cells and if you growth the cells in a serum free media the purification of a tagger protein is quite simple.
I'm using the Expi293 sistem which is expensive but really poverfull. You can just dilute the media 1:1 in a binding buffer and perform the imac purification.