@Nigel
Try cleaning the electrical contacts. Use an electrical contact cleaner, a commercial product (like CAIG DeoxIT or Stabilant-22 or a d-limonene) is best but is also very overkill, common isospropyl is a weaker solvent but it's good enough for this application ("electrical-grade" >99% isopropyl is preferred, but any drugstore ~70% isopropyl will do if it doesn't have perfumes or other additives which may leave a conductive residue). Avoid using old electrical cleaners which contain 1,1,1-trichloroethane (powerful but evil carcinogenic stuff) or which contain ketone/acetone (nail polish, etc, which can cause more damage than it fixes).
Use lint-free wipes or "microweave" cloth, Q-tips, an old (but clean) toothbrush, try to mechanically scrub the (often invisible) oxidation off the electrical contacts. On the SIM card. In the phone itself, if that's possible (which it usually isn't without a takeapart).
Even an "improper" cleaning is still better than nothing, even a "perfect" cleaning may not fix your "broken" SIM card issue (if it's "broken" from some other cause) ... but it's cheap and easy and fast and nothing to lose, worth a shot.
